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text:
Alabama woman who texted 'I feel in trouble' before she disappeared has been found dead, police say
By Amir Vera
Birmingham Police Department
Paighton Houston
SOURCE: Birmingham Police Department
Alabama police found the body of a woman missing since Dec. 20 on Friday.Paighton Houston disappeared after leaving a Birmingham bar, her family said. She texted a coworker that she was with strangers and felt she was in trouble.The body of a woman was found Friday in a shallow grave behind a house in Hueytown, Alabama, Trussville police Detective Ben Short said in a statement.The remains were positively identified as Houston's and the Jefferson County Sheriff's Office is continuing to investigate her death, Short said.Trussville police and the Metro Area Crime Center developed information Thursday that led police to the house in Hueytown, Short said.'I don't know these people,' Houston textedHouston, 29, lived in Trussville, roughly a 15-mile drive northeast of downtown Birmingham.She was visiting the Tin Roof bar in Birmingham the night she disappeared, her family says. She was last seen leaving the bar with two heavyset black men, Birmingham police say.She appeared to have left the bar willingly with the two men, police say."If I call answer, I don't know these people and I feel in trouble," the text reads, according to her mother, Charlaine Houston. Paighton Houston had driven to the bar with the coworker earlier in the day, her mother wrote.Alabama Gov. Kay Ivey offered a $5,000 reward for credible information in the search for Houston. Crime Stoppers also offered a $5,000 reward.
Alabama police found the body of a woman missing since Dec. 20 on Friday.
Paighton Houston disappeared after leaving a Birmingham bar, her family said. She texted a coworker that she was with strangers and felt she was in trouble.
The body of a woman was found Friday in a shallow grave behind a house in Hueytown, Alabama, Trussville police Detective Ben Short said in a statement.
The remains were positively identified as Houston's and the Jefferson County Sheriff's Office is continuing to investigate her death, Short said.
Trussville police and the Metro Area Crime Center developed information Thursday that led police to the house in Hueytown, Short said.
'I don't know these people,' Houston texted
Houston, 29, lived in Trussville, roughly a 15-mile drive northeast of downtown Birmingham.
She was visiting the Tin Roof bar in Birmingham the night she disappeared, her family says. She was last seen leaving the bar with two heavyset black men, Birmingham police say.
She appeared to have left the bar willingly with the two men, police say.
"If I call answer, I don't know these people and I feel in trouble," the text reads, according to her mother, Charlaine Houston. Paighton Houston had driven to the bar with the coworker earlier in the day, her mother wrote.
Alabama Gov. Kay Ivey offered a $5,000 reward for credible information in the search for Houston. Crime Stoppers also offered a $5,000 reward.

redpajama_set_name: RedPajamaCommonCrawl


text:
Surprises? I like surprises….
So… I'm back in Brisbane. I've been back a few days, and while this isn't a bad thing I'm feeling a little sad about it. Quite sad. But not too sad. There seems to have been a steady flow of little surprises of late. The Typewriters. Let's get down to business. Firstly, I managed to…
April 28, 2014 in Elron court, fixing, Olympia, Papua New Guinea., Portable, Remington, SM 9, Triumph Gabrielle E, typewriters.
Sydney Typosphere, Where the bloody hell are ya? (and your typewriters)
Steve and I woke up early again as we were eager to get into Sydney and through the traffic before it became difficult to drive on it's consistently grid-locked roads. It was a cold morning in Scone, and my car was frosted from cold condensation that had dropped overnight as it hit the nearly 30…
April 25, 2014 in Australia, collectors, having a whine, Scone. X-Trail, Sydney, Sydney Harbor Bridge, travel, typewriter, typing, Typosphere, writers, Zine.
Pushing the New England
My alarm went off at 3:30 am, and I immediately sprang to my feet (I'm a morning person, clearly). By 4:20 I was on the road to Steve Snow's place, and by 5am I had his company with me while I drove Ruby the Exxy. It was a cold easter Saturday morning, and we would…
April 22, 2014 in driving, Highway, New England, travel, typewriters, Warwick.
We're going for safari…..
April 16, 2014 in brisbane, Canberra, driving, hunting, melbourne, Sydney, typecast, typewriter, X Trail.
The blogs that could have been, never were, or are yet to be.
April 12, 2014 in 2 year birthday, Ballarat, Bendigo, Canberra, Charleville, melbourne, neglected blogs, Newcastle, Pacific Highyway., typecast, typewriters, typing.
Latest technology? Nah…. I'll stick with my Gossen.
Ohhhh…. glowing and black. Must be awesome. What you see above is an interesting little device that is referred to as a 'Robotic printer'. I spotted this Kickstater item over HERE and I found myself very curious. Check out the video on the site, and prepare to be amazed. Yeah, to be honest…
April 11, 2014 in comparison., Gossen Tippa, kickstarter, robotic printer, typewriter.
Packing typewriters like a… packing sort of person.
Words can't express how much I feel frustrated about packing this house. But a string of letters can! FFFFFFFFFFFFFAAAAAAAAAAAAAAAAAAAARRRRRRRRRRRRRRKKKKKKKKKKKKK! Anyway.. Now I've got that out, I can talk about my current attempts at packing typewriters. While the baxter boxes from work weren't large enough, I did find that I can use boxes used to transport…
April 8, 2014 in packing, typewriter love., typing, writing.
The Oz Typewriter effect and lost opportunities.
In the last year or so many of the typosphere have noted that there's been a dramatic increase in the price of typewriters sites like eBay and Etsy. While the market for these machines is growing and typewriters are no longer seen as valueless antiques sitting in the garage, there's naturally been a steady increase…
April 5, 2014 in blogs, OzTypewriter, typewriters, Typosphere.
Thoughts on 3D printing a typewriter.
It has been a while since I've addressed the question of 3D printing typewriter components. As I've so far been the only person to really attempt to tackle it, whenever the topic seems to arise in forums naturally I'll end up getting an email or a message on the subject. So I think it is…
April 4, 2014 in 3D printing, CNC, engineering, parts, production, typewriter.
There are a lot of amateur writers on the typosphere, and I include myself in that list. I think one of the things that most of us have in common is the unrelenting urge to tell a story. We have narrative in our veins that doesn't just keep us alive, but makes our hearts beat…
April 1, 2014 in Broadchurch, David Tennant, drama, grief, television, writing.


text:
Apple Music Selects Elusive R&B Singer H.E.R. For The "Up Next" Series
byYohance Kyles (@HUEYmixwitRILEY)
A short film featuring the songstress will debut November 6 exclusively on the streamer.
(AllHipHop News) Gabi "H.E.R." Wilson is on some music insiders' short list as a potential nominee for Best New Artist at the 61st Annual Grammy Awards. Apple Music is already prepared to crown the R&B singer as a notable rising musician.
The streaming giant selected H.E.R. for the "Up Next" series. The elusive vocalist follows Grammy-nominated performers such as Daniel Caesar, Khalid, and 6lack as "Up Next" alumni.
"It's an honor to be a part of Apple Music's 'Up Next' program," says H.E.R. "I'm blown away by their support in helping spread my music and my message to the world!" The multi-instrumentalist is currently the #6 R&B artist worldwide across Apple Music.
H.E.R. released the I Used to Know Her: The Prelude EP in August. Over the last year, the RCA signee earned Best New Artist nods at the 2017 Soul Train Music Awards and the 2018 BET Awards.


The West Coast town of Oban is the starting point for this delightful tour. It is easily accessible by train, coach or car and from here you cycle to Mull 'of all the isles the fairest'. After a day exploring, perhaps a wildlife tour or a trip to the peaceful Island of Iona another ferry takes you from Tobermory to Kilchoan on the remote and beautiful Ardnamurchan Peninsula.The return journey to Oban is via the popular resort of Fort William, the Isle of Lismore, famous for its wild flowers, and by ferry to the bustling town of Oban. So much variety in such a short time, the ferry crossings add to the interest of this popular tour.


There is no casting director; there is no producer monitoring your upload button. Anyone that looks like anyone can upload a video. I think YouTube and the digital space does set a really good example for the rest of the industry in that sense.
I think one of the most beautiful things about YouTube is that it makes the world a smaller place. You realize that we're all different, but we're all the same. And if you think about it, it's a beautiful concept.
When I make my own videos, I am the writer, the editor, the lighting person, everything - that's why my videos are blurry.
I know one of the reasons I first started making Youtube videos was because no one looks like me.
In my life I've gone through a lot of really hard times. I went through depression and had so many challenges that I overcame. And I overcame because I just decided to be happy.
I don't say no as much as I should. I'm an extreme workaholic. So I can be sick, and I still say yes to anything. When you are the CEO of your own company, editor of your own videos, your own writer ,and you do every role yourself, you have a hard time saying no to opportunities.
I'm my own boss, my own editor, my own shooter, my own writer, everything. This is all stuff I learned through trial and error... failing at a lot of things has taught me how to succeed at them eventually... you roll with the punches.


An exciting opportunity for researchers and business managers to explore new ways to deliver strategic priorities for a better world. Experience yourself how Europe's making life better. Take time out to consider how innovative ideas can add value in a smart, low-carbon environment. Develop skills, networks and content to disseminate your research.
Travel with the Contented team of coaches and media producers to connect with cutting-edge business ideas and practices around the world. We'll help you draw out, capture and apply new insights, and create engaging and powerful digital media and tools to increase impact.
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text:
RTR1PH4A
A poster of missing three-year-old Madeleine McCann is seen on a Police Republican Guard van in the southern...
A poster of a missing three-year-old girl is seen on a Police Republican Guard van in Algarve
A poster of missing three-year-old Madeleine McCann is seen on a Police Republican Guard van in the southern province of Algarve May 8, 2007. Police are investigating whether a paedophile ring ordered the kidnap of the three-year-old British girl who vanished from her bed five days ago in southern Portugal, daily Correio da Manha reported on Tuesday. McCann is believed to have been abducted from her apartment on May 3 in Praia da Luz, a popular resort in the Algarve, just yards away from where her parents were dining. REUTERS/Hugo Correia (PORTUGAL)
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Hugo Correia
McCANN Madeleine


A beautiful introduction to your celebration, these unique graduation announcements feature strands of confetti in your choice of three shimmering foil colors on the front with your graduation details. The back features your choice of background color and a photo collage of 20 photos! Your wording is printed in your choice of colors and fonts.


text:
\section{Introduction}

The study of phylogenetic networks as a model of reticulate evolution
began with the representation of conflicting phylogenetic signals as
an implicit \emph{splits
nework}~\cite{bandelt.dress:1992,huson.bryant:2006}, but it was soon
realized that internal nodes in a splits network did not have any
direct interpretation in evolutionary terms.  Attention turned then to
the study of explicit evolutionary networks, in which the internal
nodes have a direct interpretation as reticulate evolutionary events
such as recombination, hybridization, or lateral gene transfer. Unfortunately,
the hardness of reconstructing an evolutionary network with as few
recombination events as possible for a set of sequences, under the
assumption of no repeated or back mutations, was soon
established~\cite{bordewich.sample:2004,wang.ea:2000,wang.ea:2001}.

However, when the conflicting phylogenetic signals show a particular structure,
such that the \emph{conflict graph} of the set of sequences is
\emph{biconvex}, the evolutionary network with the smallest
possible number of recombination events
is unique, it can be reconstructed in polynomial time and it is a \emph{galled tree},
an evolutionary network with hybrid nodes of in-degree 2 (because they correspond to explicit recombinations) and disjoint reticulation cycles~\cite{gusfield.ea:csb:2003}.
Galled trees are also relevant from a biological point of view because, as Gusfield \textsl{et al} point out in \textsl{loc. cit.}, reticulation events tend to be isolated, yielding to disjoint reticulation cycles,  if the level of recombination is moderate, or
if most of the observable recombinations are recent.
Actually, several slightly different notions of galled tree have been introduced so far in the literature, depending on the degree of disjointness of their reticulation cycles.
The original galled trees \cite{gusfield.ea:csb:2003} have node-disjoint reticulation cycles, while the \emph{nested networks with nesting level 1}~\cite{JanssonSung2004a,JanssonSung2008} (dubbed, for simplicity, \emph{1-nested networks} in this paper) have arc-disjoint  reticulation cycles. Between both notions lie the level-1 networks~\cite{CJSS2005,jansson.sung:2006}, without biconnected components with more than one hybrid node.  We have studied the relationships among these types of networks~\cite{rv:galled09}: see Section~\ref{sec:gt} below.

Now, various algorithms are known for reconstructing galled trees from
either
sequences~\cite{gusfield.ea:2007,gusfield.ea:csb:2003,gusfield.ea:informs:2004,gusfield.ea:2004},
trees~\cite{nakhleh.ea:2005}, distances~\cite{chan.ea:2006},
splits~\cite{huson.kloepper:2007}, or triplets~\cite{jansson.ea:2006},
and metrics provide a safe way to assess phylogenetic reconstruction
methods~\cite{miyamoto.cracraft:1991,nakhleh.ea:03psb}. A few
polynomial time computable metrics, like the path multiplicity or
$\mu$-distance \cite{cardona.ea:07b} or Nakhleh's metric $m$ for reduced networks \cite{nakhleh:2009},
 are known for \emph{tree child} evolutionary networks~\cite{cardona.ea:07b}, which include galled trees \cite{rv:galled09}. But most distance measures introduced so far were only known to be metrics on \emph{time consistent}~\cite{baroni.ea:2006}
 tree-child phylogenetic
networks, including the Robinson-Foulds
distance~\cite{baroni.ea:ac04,cardona.ea:07a,cardona.ea:tcbb:comparison.1:2008}, the tripartitions
distance~\cite{cardona.ea:07a,moret.ea:2004},
the nodal and splitted nodal
distances~\cite{nodalnetworks,cardona.ea:tcbb:comparison.2:2008}, and
the triplets distance~\cite{cardona.ea:tcbb:comparison.2:2008}.
Since galled trees need not be time consistent, it was not known whether these distance measures define metrics for galled trees. On the other hand,  Nakhleh gave
in his PhD Thesis \cite{nakhleh:phd04} two metrics for time consistent galled trees
(based on splits and subtrees), but they are not metrics for arbitrary galled
trees \cite{cardona.ea:07b}. Recent simulation studies using the
coalescent model with recombination show that only a small fraction of
the simulated galled trees are time consistent~\cite{arenas.ea:2008}.

In this paper, we study which of the aforementioned metrics for tree-child time consistent phylogenetic networks are also metrics for galled trees, under the various notions of the latter. We show that the Robinson-Foulds distance is only a metric in the binary case (in which the original galled trees, the level-1 networks and the 1-nested networks are the same objects); the tripartitions distance is a metric for 1-nested networks  without any restriction on the degrees of their nodes (besides the general restriction that hybrid nodes have in-degree 2); and the splitted nodal distance is a metric in the \emph{semibinary} (hybrid nodes of in-degree 2 and out-degree 1) case, in which the 1-nested and level-1 conditions define the same objects, but they are strictly weaker than the node-disjoint reticulation cycles condition). On the other hand, neither the nodal distance nor the triplets distance are metrics even for the most restrictive case of binary galled trees.

\section{Preliminaries}
\label{sec:prel}

Given a set $S$, a \emph{$S$-rDAG} is a rooted directed acyclic graph
with its leaves bijectively labeled in $S$.

A \emph{tree node} of a $S$-rDAG $N=(V,E)$ is a node of in-degree at
most 1, and a \emph{hybrid node} is a node of in-degree at least 2.  A
\emph{tree arc} (respectively, a \emph{hybridization arc}) is an arc
with head a tree node (respectively, a hybrid node).  A node $v\in V$
is a \emph{child} of $u\in V$ if $(u,v)\in E$; we also say in this
case that $u$ is a \emph{parent} of $v$.  Two nodes are \emph{sibling}
when they have a common parent.

We denote by $u\!\rightsquigarrow\!{} v$ any path in $N$ with origin $u$ and end $v$.
Whenever there exists a path $u\!\rightsquigarrow\!{} v$, we shall say that $v$ is a
\emph{descendant} of $u$ and also that $u$ is an \emph{ancestor} of
$v$.  The \emph{length} of a path is its number of arcs, and the
\emph{distance} from a node $u$ to a descendant $v$ of it is the
length of a shortest path $u\!\rightsquigarrow\!{} v$.

A node $v$ is a \emph{strict} descendant of a node $u$ in $N$ when
every path from the root of $N$ to $v$ contains the node $u$; thus,
$v$ is a \emph{non-strict} descendant of $u$ when it is a descendant
of $u$, but there exist paths from the root to $v$ that do not contain
$u$.  The following straightforward result, which is Lemma 1 in
\cite{cardona.ea:tcbb:comparison.1:2008}, will be used often, usually
without any further notice.

\begin{lemma}
\label{lem:strict->anc}
Every strict ancestor of a node $v$ is connected by a path with every
ancestor of $v$. \qed
\end{lemma}

A \emph{tree path} is a path consisting only of tree arcs, and a
node $v$ is a \emph{tree descendant} of a node $u$ when there is a
tree path $u\!\rightsquigarrow\!{} v$.  The following result summarizes Lemma 3 and
Corollary 4 in \cite{cardona.ea:07a}, and it will also be used many
times in this paper without
any further notice.

\begin{lemma}\label{lem:unicity-path}
Let $u\!\rightsquigarrow\!{} v$ be a tree path in a $S$-rDAG.
\begin{enumerate}[(1)]

\item Every other path $w\!\rightsquigarrow\!{} v$ ending in $v$  either is
contained in $u\!\rightsquigarrow\!{} v$ or contains $u\!\rightsquigarrow\!{} v$.  In particular,
if $w$ is a descendant of $u$ and there exists a path $w\!\rightsquigarrow\!{} v$,
then this path is contained in the tree path $u\!\rightsquigarrow\!{} v$.

\item The tree path $u\!\rightsquigarrow\!{} v$ is the unique path from $u$ to $v$.

\item The node $v$ is a strict descendant of $u$. \qed
\end{enumerate}
\end{lemma}

Two paths in a $S$-rDAG are \emph{internally disjoint} when they have
disjoint sets of intermediate nodes.  A \emph{reticulation cycle} for
a hybrid node $h$ is a pair of internally disjoint paths ending in $h$
and with the same origin.  Each one of the paths forming a
reticulation cycle for $h$ is called generically a \emph{merge path},
their common origin is called the \emph{split node} of the
reticulation cycle, and the hybrid node $h$, its \emph{end}.  The
\emph{intermediate nodes} of a reticulation cycle are the intermediate
nodes of the merge paths forming it.

A subgraph of an undirected graph is \emph{biconnected} when it is
connected and it remains connected if we remove any node and all edges
incident to it.  A subgraph of a $S$-rDAG $N$ is said to be
\emph{biconnected} when it is so in the undirected graph associated to
$N$.

\section{1-nested networks}
\label{sec:gt}

In the rest of this paper, by a \emph{hybridization network} on a set
$S$ we understand a $S$-rDAG without out-degree 1 tree nodes and with
all its hybrid nodes of in-degree 2.  We shall also use the term
\emph{hybridization network with $n$ leaves} to refer to a
hybridization network on a set $S$ with $n$ elements.  A
\emph{phylogenetic tree} is a hybridization network without hybrid
nodes.

We shall say that a hybridization network is \emph{semibinary} when
its hybrid nodes have out-degree 1, and that it is \emph{binary} when
it is semibinary and its internal tree nodes have out-degree 2.

A hybridization network is:
\begin{itemize}

\item a \emph{galled tree}, when every pair of reticulation cycles
have disjoint sets of nodes \cite{gusfield.ea:csb:2003}.

\item \emph{1-nested}, when every pair of reticulation cycles have
disjoint sets of arcs: by \cite[Prop.~12]{rv:galled09}, this is
equivalent to the fact that every pair of reticulation cycles for
different hybrid nodes have disjoint sets of intermediate nodes, and
hence it corresponds to the notion of \emph{nested (hybridization)
network with nesting depth 1} \cite{JanssonSung2004a,JanssonSung2008}.

\item \emph{level-1}, when no biconnected component contains more than
1 hybrid node \cite{CJSS2005,jansson.sung:2006}.
\end{itemize}
To simplify the language, from now on we shall write simply
\emph{1-nested network} to mean a 1-nested hybridization network.  The
following two results summarize the main results on 1-nested networks
proved in \cite{rv:galled09}.

\begin{lemma}\label{1n no ih}
In a 1-nested network, every hybrid node is the end of exactly one
reticulation cycle, and all the intermediate nodes of this
reticulation cycle are of tree type.  \qed
\end{lemma}

\begin{theorem}\label{resum}
\begin{enumerate}[(a)]

\item Every 1-nested network is \emph{tree-child}, in the sense that
every internal node has a child of tree type.

\item For general hybridization networks,
$$
\mbox{galled tree}\Longrightarrow \mbox{level-1} \Longrightarrow
\mbox{1-nested},
$$
and these implications are strict.

\item For semibinary hybridization networks,
$$
\mbox{galled tree}\Longrightarrow \mbox{level-1} \Longleftrightarrow
\mbox{1-nested},
$$
and the first implication is strict.

\item For binary hybridization networks,
$$
\mbox{galled tree} \Longleftrightarrow \mbox{level-1}
\Longleftrightarrow \mbox{1-nested. \qed}
$$
\end{enumerate}
\end{theorem}

The fact that every 1-nested network is tree-child implies, by
\cite[Lem.~2]{cardona.ea:07a} the following result.

\begin{corollary}\label{lem:1n->tc}
Every node in a 1-nested   network has some tree
descendant leaf, and hence some strict descendant leaf. \qed
\end{corollary}

The following result lies at the basis of most of our proofs.

\begin{proposition}\label{prop:basic-2h1n}
Every 1-nested   network contains some internal tree node
with all its children tree leaves, or a hybrid node with all its
children tree leaves and such that all the intermediate nodes in its
reticulation cycle have all their children outside the reticulation
cycle tree leaves.
\end{proposition}

\begin{proof}
Let $N$ be a 1-nested network.  Let the \emph{galled-length} of a path
in $N$ be the number of reticulation cycles which the arcs of the path
belong to, and the \emph{galled-depth} of a node in $N$ the largest
galled-length of a path from the root to it.  Notice that the
galled-depth of a hybrid node is equal to the galled-depth of the
intermediate nodes of its reticulation cycle (because every arc in $N$
belongs at most to one reticulation cycle).

Assume that $N$ does not contain any internal tree node with all its
children tree leaves.  Let $h$ be a hybrid node of largest
galled-depth in $N$, and let $v$ denote either $h$ or any intermediate
node in the reticulation cycle $K$ for $h$.  It turns out that $v$ has
no hybrid descendant other than $h$, because any path from $v$ to any
other hybrid node $h'\neq h$ would contain arcs belonging to at least
one more reticulation cycle, making the galled-depth of $h'$ larger
than that of $v$.

Let $v'$ be any descendant of $v$ not belonging to $K$.  Then, $v'$ is
a tree node and all its descendants are tree nodes, and therefore,
since we assume that $N$ does not contain any internal tree node with
all its children tree leaves, we conclude that $v'$ is a tree leaf.
\end{proof}

\section{Reductions for 1-nested networks}
\label{sec:red-1n}

We introduce in this section a set of reductions for 1-nested
networks.  Each of these reductions, when applied to a 1-nested
network with $n$ leaves and $m$ nodes, produces a 1-nested network
with at most $n$ leaves and less than $m$ nodes, and given any
1-nested network with more than one leaf, it is always possible to
apply to it some of these reductions.  We shall also show that
suitable subsets of these reductions have similar properties for
binary and for semibinary 1-nested networks.  Similar sets of
reductions for other types of evolutionary networks have already been
published
\cite{cardona.ea:sbtstc:2008,cardona.ea:tcbb:comparison.2:2008}.
\smallskip

\noindent\textbf{\emph{The $R$ reductions.}} Let $N$ be a 1-nested
network with $n$ leaves, and let $u$ be an internal node whose
children are exactly the tree leaves $i$ and $j$.  The \emph{$R_{i;j}$
reduction} of $N$ is the network $R_{i;j}(N)$ obtained by removing the
leaves $i$ and $j$, together with their incoming arcs, and labeling
with $i$ their former common parent $u$, which has become now a leaf;
cf.~Fig.~\ref{fig:R-red}.\footnote{In graphical representations of
hybridization networks, we shall represent hybrid nodes by squares,
tree nodes by circles, and indeterminate (that is, that can be of tree
or hybrid type) nodes by pentagons.} It is clear that $R_{i;j}(N)$ is
a 1-nested network on $S\setminus\{j\}$, and it has 2 nodes less than
$N$.  \smallskip

\begin{figure}[htb]
\centering
  \begin{tikzpicture}[thick,>=stealth,scale=0.3
    \draw (0,2) node[tri] (xx) {};
    \draw (0,0) node[tri] (u) {};
    \draw (-1,-2) node[tre] (i) {}; \etq {i}
    \draw (1,-2) node[tre] (j) {}; \etq {j}
    \draw [->](xx)--(u);
    \draw [->](u)--(i);
    \draw [->](u)--(j);
  \end{tikzpicture}
  \qquad
               \begin{tikzpicture}[thick,>=stealth,scale=0.3]
              \draw(0,-2) node{\ };
              \draw(0,0) node  {$\Rightarrow$};
              \draw(0,2) node{\ };
 \end{tikzpicture}
  \qquad
  \begin{tikzpicture}[thick,>=stealth,scale=0.3
    \draw (0,2) node[tri] (xx) {};
    \draw (0,0) node[tri] (u) {}; \draw (u) node {\tiny $i$};
    \draw (0,-2) node  {\ };
    \draw [->](xx)--(u);
  \end{tikzpicture}

\caption{\label{fig:R-red}
The $R_{i;j}$ reduction.}
\end{figure}

\noindent\textbf{\emph{The $T$ reductions.}} Let $N$ be a 1-nested
network with $n$ leaves, and let $u$ be an internal node with two tree
leaf children $i,j$ and at least some other child.  The
\emph{$T_{i;j}$ reduction} of $N$ is the network $T_{i;j}(N)$ obtained
by removing the leaf $j$ together with its incoming arc;
cf.~Fig.~\ref{fig:T-red-1}.  It is clear that $T_{i;j}(N)$ is a
1-nested network on $S\setminus\{j\}$ with 1 node less than $N$.
\smallskip

\begin{figure}[htb]
\centering
  \begin{tikzpicture}[thick,>=stealth,scale=0.3
    \draw (0,2) node[tri] (xx) {};
    \draw (0,0) node[tri] (u) {}; \etq{u}
    \draw (-3,-2) node[tre] (j) {}; \etq {j}
    \draw (-1,-2) node[tre] (i) {};  \etq {i}
       \draw (1,-2) node  {$\ldots$};
    \draw (3,-2) node[tri] (ik) {};
 \draw [->](xx)--(u);
    \draw [->](u)--(i);
      \draw [->](u)--(j);
    \draw [->](u)--(ik);
  \end{tikzpicture}
  \qquad
               \begin{tikzpicture}[thick,>=stealth,scale=0.3]
              \draw(0,-2) node{\ };
              \draw(0,0) node  {$\Rightarrow$};
              \draw(0,2) node{\ };
 \end{tikzpicture}
  \qquad
  \begin{tikzpicture}[thick,>=stealth,scale=0.3
     \draw (0,2) node[tri] (xx) {};
    \draw (0,0) node[tri] (u) {}; \etq{u}
    \draw (-1.5,-2) node[tre] (i) {};  \etq {i}
       \draw (0,-2) node  {$\ldots$};
    \draw (1.5,-2) node[tri] (ik) {};
 \draw [->](xx)--(u);
  \draw [->](u)--(i);
    \draw [->](u)--(ik);
  \end{tikzpicture}

\caption{\label{fig:T-red-1}
The $T_{i;j}$ reduction.}
\end{figure}

\noindent\textbf{\emph{The $G$ reductions.}} Let $N$ be a 1-nested
network with $n$ leaves.  Assume that $N$ contains a reticulation
cycle $K$ consisting of two merge paths $(u,v_1,\ldots,v_k,h)$ and
$(u,v'_1,\ldots,v'_{k'},h)$, with $k\geq k'$ ($k'$ can be 0, in which
case the corresponding merge path is simply the arc $(u,h)$, but then
$k>0$), such that
\begin{itemize}

\item the hybrid node $h$ has only one child, and it is the tree leaf
$i$;

\item each intermediate node of $K$ has only one child outside $K$,
and it is a tree leaf: the child outside $K$ of each $v_j$ is the leaf
$i_j$ and the child outside $K$ of each $v'_j$ is the leaf $i_j'$.
\end{itemize}
Notice that $u$ may have children outside $K$.

The \emph{$G_{i;i_1,\dots,i_k;i'_1,\dots,i'_{k'}}$ reduction} of $N$
is the network $G_{i;i_1,\dots,i_k;i'_1,\dots,i'_{k'}}(N)$ obtained by
removing the nodes $v_1,\ldots,v_k,v_1',\ldots,v_k',h$ and the leaves
$i_1,\ldots,i_k,i'_1,\ldots,i'_{k'},i$, together with all their
incoming arcs, and then adding to the node $u$ two new tree leaf
children, labeled $i$ and $i_1$; cf.~Fig.~\ref{fig:G-red}.  Since we
remove a complete reticulation cycle and all descendants of its
intermediate nodes, and we replace it by two tree leaves, it is clear
that $G_{i;i_1,\dots,i_k;i'_1,\dots,i'_{k'}}(N)$ is a 1-nested network
on $S\setminus\{i_2,\dots,i_k,i'_1,\dots,i'_{k'}\}$ (and in
particular, if $k=1$ and $k'=0$, it has the same leaves as $N$) with
$2(k+k')$ nodes less than $N$.

\begin{figure}[htb]
\centering
  \begin{tikzpicture}[thick,>=stealth,scale=0.3
    \draw (0,2) node[tri] (xx) {};
    \draw (0,0) node[tri] (u) {}; \etq{u}
       \draw (5,-2) node[tri] (v) {};
  \draw (6.5,-2) node  {$\ldots$};
    \draw (-2,-2) node[tre] (v_1) {}; \etq{v_1}
    \draw (-2,-4) node[tre] (v_2) {}; \etq{v_2}
    \draw (-2,-5) node  {$\vdots$};
    \draw (-2,-6.5) node[tre] (v_k) {}; \etq{v_k}
    \draw (0,-8.5) node[hyb] (h) {}; \etq{h}
 \draw (2,-2) node[tre] (v'_1) {}; \etq{v'_1}
    \draw (2,-4) node[tre] (v'_2) {}; \etq{v'_2}
    \draw (2,-5) node  {$\vdots$};
    \draw (2,-6.5) node[tre] (v'_{k'}) {}; \etq{v'_{k'}}
   \draw (-4,-3) node[tre] (i_1) {}; \etq{i_1}
    \draw (-4,-5) node[tre] (i_2) {}; \etq{i_2}
    \draw (-4,-7.5) node[tre] (i_k) {}; \etq{i_k}
    \draw (0,-10.5) node[tre] (i) {}; \etq{i}
 \draw (4,-3) node[tre] (i'_1) {}; \etq{i'_1}
    \draw (4,-5) node[tre] (i'_2) {}; \etq{i'_2}
    \draw (4,-7.5) node[tre] (i'_{k'}) {}; \etq{i'_{k'}}
          \draw[->](u) to (v);
  \draw[->](xx)--(u);
    \draw[->](u) to (v_1);
        \draw[->](u) to (v'_1);
    \draw[->](v_1) to (i_1);
    \draw[->](v_1) to (v_2);
    \draw[->](v_2) to (i_2);
    \draw[->](v_k) to (i_k);
    \draw[->](v_k) to (h);
    \draw[->](v'_1) to (i'_1);
    \draw[->](v'_1) to (v'_2);
     \draw[->](v'_2) to (i'_2);
   \draw[->](v'_{k'}) to (i'_{k'});
    \draw[->](v'_{k'}) to (h);
    \draw[->](h) to (i);
  \end{tikzpicture}
               \begin{tikzpicture}[thick,>=stealth,scale=0.3]
              \draw(0,2) node{\ };
              \draw(0,0) node  {$\Longrightarrow$};
              \draw(0,-10.5) node{\ };
 \end{tikzpicture}
  \begin{tikzpicture}[thick,>=stealth,scale=0.3
     \draw (0,2) node[tri] (xx) {};
    \draw (0,0) node[tri] (u) {}; \etq{u}
    \draw (-2,-2) node[tre] (i_1) {}; \etq{i_1}
       \draw (0,-2) node[tre] (i) {}; \etq{i}
       \draw (2,-2) node[tri] (v) {};
       \draw (3.5,-2) node  {$\ldots$};
  \draw (0,-10.5) node  {\ };
   \draw[->](xx) to (u);
      \draw[->](u) to (i);
      \draw[->](u) to (i_1);
      \draw[->](u) to (v);

     \end{tikzpicture}

\caption{\label{fig:G-red}
The $G_{i;i_1,\dots,i_k;i'_1,\dots,i'_{k'}}$ reduction.}
\end{figure}

\noindent\textbf{\emph{The $\overline{G} $ reductions.}} The
$\overline{G}_{i;i_1,\ldots,i_k; i'_1,\ldots,i'_{k'}}$ reduction is the same
as $G_{i;i_1,\dots,i_k;i'_1,\dots,i'_{k'}}$, except for the fact that
in order to apply the $\overline{G}_{i;i_1,\ldots,i_k; i'_1,\ldots,i'_{k'}}$
reduction, the hybrid node $h$ must be the leaf labeled $i$, instead
of the leaf's parent: see~Fig.~\ref{fig:G-red-1}.  Then,
$\overline{G}_{i;i_1,\dots,i_k;i'_1,\dots,i'_{k'}}(N)$ is a 1-nested network
on $S\setminus\{i_2,\dots,i_k,i'_1,\dots,i'_{k'}\}$ and it has
$2(k+k')-1$ nodes less than $N$.

\begin{figure}[htb]
\centering
  \begin{tikzpicture}[thick,>=stealth,scale=0.3
    \draw (0,2) node[tri] (xx) {};
    \draw (0,0) node[tri] (u) {}; \etq{u}
     \draw (5,-2) node[tri] (v) {};
  \draw (6.5,-2) node  {$\ldots$};
    \draw (-2,-2) node[tre] (v_1) {}; \etq{v_1}
    \draw (-2,-4) node[tre] (v_2) {}; \etq{v_2}
    \draw (-2,-5) node  {$\vdots$};
    \draw (-2,-6.5) node[tre] (v_k) {}; \etq{v_k}
    \draw (0,-8.5) node[hyb] (i) {}; \etq{i}
 \draw (2,-2) node[tre] (v'_1) {}; \etq{v'_1}
    \draw (2,-4) node[tre] (v'_2) {}; \etq{v'_2}
    \draw (2,-5) node  {$\vdots$};
    \draw (2,-6.5) node[tre] (v'_{k'}) {}; \etq{v'_{k'}}
   \draw (-4,-3) node[tre] (i_1) {}; \etq{i_1}
    \draw (-4,-5) node[tre] (i_2) {}; \etq{i_2}
    \draw (-4,-7.5) node[tre] (i_k) {}; \etq{i_k}
 \draw (4,-3) node[tre] (i'_1) {}; \etq{i'_1}
    \draw (4,-5) node[tre] (i'_2) {}; \etq{i'_2}
    \draw (4,-7.5) node[tre] (i'_{k'}) {}; \etq{i'_{k'}}
    \draw[->](xx)--(u);
        \draw[->](u) to (v);
    \draw[->](u) to (v_1);
        \draw[->](u) to (v'_1);
    \draw[->](v_1) to (i_1);
    \draw[->](v_1) to (v_2);
    \draw[->](v_2) to (i_2);
    \draw[->](v_k) to (i_k);
    \draw[->](v_k) to (i);
    \draw[->](v'_1) to (i'_1);
    \draw[->](v'_1) to (v'_2);
     \draw[->](v'_2) to (i'_2);
   \draw[->](v'_{k'}) to (i'_{k'});
    \draw[->](v'_{k'}) to (i);
  \end{tikzpicture}
               \begin{tikzpicture}[thick,>=stealth,scale=0.3]
              \draw(0,2) node{\ };
              \draw(0,0) node  {$\Longrightarrow$};
              \draw(0,-8.5) node{\ };
 \end{tikzpicture}
  \begin{tikzpicture}[thick,>=stealth,scale=0.3
     \draw (0,2) node[tri] (xx) {};
    \draw (0,0) node[tri] (u) {}; \etq{u}
    \draw (-2,-2) node[tre] (i_1) {}; \etq{i_1}
       \draw (0,-2) node[tre] (i) {}; \etq{i}
       \draw (2,-2) node[tri] (v) {};
       \draw (3.5,-2) node  {$\ldots$};
  \draw (0,-8.5) node  {\ };
   \draw[->](xx) to (u);
      \draw[->](u) to (i);
      \draw[->](u) to (i_1);
      \draw[->](u) to (v);

     \end{tikzpicture}

\caption{\label{fig:G-red-1}
The $\overline{G}_{i;i_1,\dots,i_k;i'_1,\dots,i'_{k'}}$ reduction.}
\end{figure}

\begin{remark}
In the $G$ and $\overline{G}$ reductions, we leave two tree leaves attached to
the former split node of the removed reticulation cycle in order to
ensure that their application never generates an out-degree 1 tree
node, while avoiding to increase unnecessarily the number of
reductions.
\end{remark}

Now we have the following basic applicability results.

\begin{proposition}\label{thm:red}
Let $N$ be a 1-nested network with more than one leaf.  Then, at least
one $R$, $T$, $G$, or $\overline{G}$ reduction can be applied to $N$, and
the result is a 1-nested network.
\end{proposition}

\begin{proof}
If $N$ contains some internal node $v$ with at least two children that
are tree leaves, say $i$ and $j$, then we can apply to $N$ the
$R_{i;j}$ reduction, if the out-degree of $v$ is 2, or the $T_{i;j}$
reduction, if its out-degree is greater than 2.

Assume now that $N$ does not contain any internal node with more than
one tree leaf child: in particular, it does not contain any internal
tree node with all its children tree leaves.  Then, by Proposition
\ref{prop:basic-2h1n}, it contains a hybrid node $h$ with all its
children tree leaves (and therefore, by the current assumption on $N$,
$h$ either is a leaf itself or has out-degree 1), and such that all
the intermediate nodes in its reticulation cycle $K$ have all their
children outside $K$ tree leaves (and therefore each one of them has
exactly one child outside $K$, by Lemma \ref{1n no ih} and the current
assumption on $N$): let $i_1,\ldots, i_k$ and $i'_1, \ldots, i'_{k'}$,
with $k\geq k'$, be the tree leaf children of the intermediate nodes
of the two respective merge paths of $K$, listed in descending order
of their parents along the path.  Then, if $h$ has out-degree 1 and
its child is the tree leaf $i$, we can apply to $N$ the
$G_{i;i_1,\ldots,i_k;i_1',\ldots,i_{k'}'}$ reduction, while if $h$ is
the leaf $i$, we can apply to $N$ the
$\overline{G}_{i;i_1,\ldots,i_k;i_1',\ldots,i_{k'}'}$ reduction.

The fact that the result of the application of a $R$, $T$, $G$, or
$\overline{G}$ reduction to $N$ is again a 1-nested network has been discussed
in the definition of the reductions.
\end{proof}

\begin{corollary}\label{thm:red-sbin}
Let $N$ be a semibinary 1-nested network with more than one leaf.
Then, at least one $R$, $T$, or $G$ reduction can be applied to $N$,
and the result is a semibinary 1-nested network.
\end{corollary}

\begin{proof}
Since $N$ does not contain hybrid leaves, we cannot apply to it any
$\overline{G}$ reduction, and therefore, by Proposition \ref{thm:red}, we can
apply to it at least one $R$, $T$, or $G$ reduction.

Now, if we can apply a $R_{i;j}$ or $T_{i;j}$ reduction to $N$, the
common parent of the tree leaves $i$ and $j$ is a tree node, and if we
can apply a $G_{i;i_1,\ldots,i_k;i'_1,\ldots,i'_{k'}}$ reduction, the
split node of the reticulation cycle for the hybrid parent of $i$ is a
tree node (in both cases because hybrid nodes in $N$ have out-degree
1), and therefore neither application produces a hybrid node of
out-degree different from 1.
\end{proof}

\begin{corollary}\label{thm:red-bin}
Let $N$ be a binary 1-nested network with more than one leaf.  Then,
at least one $R$ or $G$ reduction can be applied to $N$, and the
result is a binary 1-nested network.
\end{corollary}

\begin{proof}
Since $N$ does not contain nodes with out-degree greater than 2, we
cannot apply to it any $T$ reduction, and thus, by Corollary
\ref{thm:red-sbin}, we can apply to it some $R$ or $G$ reduction.

Now, if we apply a $R$ reduction to $N$, we replace an internal tree
node with two tree leaf children by a tree leaf, and the result is
again binary.  And if we apply to $N$ a $G$ reduction, the split node
of the reticulation cycle we remove is, as in the semibinary case, a
tree node, and in this case moreover without any child outside the
reticulation cycle (because its out-degree must be 2), and after the
application of the reduction it is still a tree node of out-degree 2.
\end{proof}

We shall call the inverses of the $R$, $T$, $G$, and $\overline{G}$
reductions, respectively, the $R^{-1}$, $T^{-1}$, $G^{-1}$, and
$\overline{G}^{-1}$ \emph{expansions}, and we shall denote them by
$R_{i;j}^{-1}$, $T^{-1}_{i;j}$,
$G_{i;i_1,\ldots,i_k;i_1',\ldots,i_{k'}'}^{-1}$, and
$\overline{G}_{i;i_1,\ldots,i_k;i_1',\ldots,i_{k'}'}^{-1}$.  More
specifically, for every 1-nested network $N$:
\begin{itemize}

\item if $N$ contains a leaf labeled with $i$ but no leaf labeled with
$j$, then the $R_{i;j}^{-1}$ expansion can be applied to $N$, and
$R_{i;j}^{-1}(N)$ is obtained by unlabeling the leaf $i$ and adding to
it two tree leaf children labeled $i$ and $j$;

\item if $N$ contains a tree leaf labeled with $i$ that has some
sibling, but no leaf labeled with $j$, then the $T_{i;j}^{-1}$
expansion can be applied to $N$, and $T_{i;j}^{-1}(N)$ is obtained by
adding to the parent of the leaf $i$ a new tree leaf child labeled
with $j$;

\item if $N$ contains an internal node $u$ with two tree leaf children
$i_1,i$, but no leaf labeled with $i_2,\ldots,i_k,i_1',\ldots,i'_{k'}$
(with $k\geq k'$), then the
$G_{i;i_1,\ldots,i_k;i_1',\ldots,i_{k'}'}^{-1}$ expansion can be
applied to $N$, and $G_{i;i_1,\ldots,i_k;i_1',\ldots,i_{k'}'}^{-1}(N)$
is obtained by removing the leaves $i,i_1$ and their incoming arcs,
and then starting in $u$ two new internally disjoint paths with $k$
and $k'$, respectively, intermediate nodes and ending in the same
hybrid node $h$, and then adding to each intermediate node of these
paths one new tree leaf and labeling these leaves (in descending order
along the paths) with $i_1,\ldots,i_k$ and $i_1',\ldots,i'_{k'}$,
respectively, and finally adding to $h$ a new tree leaf child labeled
with $i$;

\item the application condition for the
$\overline{G}_{i;i_1,\ldots,i_k;i_1',\ldots,i_{k'}'}^{-1}$ expansion is
exactly the same as for
$G_{i;i_1,\ldots,i_k;i_1',\ldots,i_{k'}'}^{-1}$, and
$\overline{G}_{i;i_1,\ldots,i_k;i_1',\ldots,i_{k'}'}^{-1}(N)$ is as
$G_{i;i_1,\ldots,i_k;i_1',\ldots,i_{k'}'}^{-1}(N)$, except that the
new hybrid node is itself a leaf labeled with $i$.
\end{itemize}
>From these descriptions we easily see that the result of a $R^{-1}$,
$T^{-1}$, $G^{-1}$ or $\overline{G}^{-1}$ expansion applied to a 1-nested
network is always a 1-nested network.

The following result is easily deduced from the explicit descriptions
of the reductions and expansions.

\begin{lemma}
\label{lem:iso-red}
Let $N$ and $N'$ be two 1-nested networks.  If $N\cong N'$, then the
result of applying to both $N$ and $N'$ the same $R^{-1}$ expansion
(respectively, $T^{-1}$ expansion, $G^{-1}$ expansion or $\overline{G}^{-1}$
expansion) are again two isomorphic 1-nested networks.

Moreover, if we apply a $R$, $T$, $G$, or $\overline{G}$ reduction to a
1-nested network $N$, then we can apply to the resulting network the
corresponding inverse $R^{-1}$, $T^{-1}$, $G^{-1}$, or $\overline{G}^{-1}$
expansion and the result is a 1-nested network isomorphic to $N$.
\qed
\end{lemma}

\section{Proving metrics through reductions}
\label{sec:proving}

Let $\mathcal{C}$ be throughout this section a class endowed with a
notion of isomorphism $\cong$.  A \emph{metric} on $\mathcal{C}$ is a
mapping
$$
d:\mathcal{C}\times \mathcal{C} \to \mathbb{R}
$$
satisfying the following axioms: for every $A,B,C\in \mathcal{C}$,
\begin{enumerate}[(a)]

\item \emph{Non-negativity}: $d(A,B)\geq 0$;

\item \emph{Separation}: $d(A,B)=0$ if and only if $A\cong B$;

\item \emph{Symmetry}: $d(A,B)=d(B,A)$;

\item \emph{Triangle inequality}: $d(A,C)\leq d(A,B)+d(B,C)$.
\end{enumerate}
A \emph{metric space} is a pair $(X,d)$ where $X$ is a set and $d$ is
a metric on $X$, taking as the notion of isomorphism in $X$ the
equality (that is, replacing $\cong$ by $=$ in the separation axiom).

All distances for hybridization networks considered in this paper are
\emph{induced through representations}, in the following sense.  A
\emph{representation} of $\mathcal{C}$ in a metric space $(X,d)$ is a
mapping
$$
F:\mathcal{C}\to X
$$
such that if $A\cong B$, then $F(A)=F(B)$.

Given such a representation, the \emph{distance induced} by $d$
through $F$ is the mapping
$$
d_F:\mathcal{C}\times\mathcal{C}\to \mathbb{R}
$$
defined by $d_F(A,B)=d(F(A),F(B))$, for every $A,B\in \mathcal{C}$.

The metric axioms for $d$ imply that this mapping is non-negative,
symmetric, it sends pairs of isomorphic members of $\mathcal{C}$ to 0, and it
satisfies the triangle inequality.  So, to be a metric on $\mathcal{C}$, $d_F$
only needs to satisfy that $d_F(A,B)=0$ implies $A\cong B$.  Now, it
is straightforward to prove the following result (cf.
\cite[Prop.~1]{cardona.ea:09-nak}).

\begin{lemma}\label{prop:metrics}
The mapping $d_F$ is a metric on $\mathcal{C}$ if, and only if, it is
injective up to isomorphism, in the sense that, for every $A,B\in
\mathcal{C}$, if $F(A)=F(B)$, then $A\cong B$.  \qed
\end{lemma}

Reductions as those introduced in the last section can be used to
prove the injectivity up to isomorphism of a representation $F$ and
hence, as a consequence, that the corresponding $d_F$ is a metric; it
was done for specific classes $\mathcal{C}$ of evolutionary networks
and specific metrics in
\cite{cardona.ea:sbtstc:2008,cardona.ea:tcbb:comparison.2:2008}.
Since we shall use several times this kind of proofs in this paper, we
make explicit here their general outline and the lemma they rely on.

Let $\mathcal{C}_{S',m}$ denote a class of 1-nested hybridization
networks of some specific type on a given set $S'$ and with at most
$m$ nodes, and let $\mathcal{C}_{S'}=\bigcup_{m\geq |S'|}
\mathcal{C}_{S',m}$.  Assume we have a set of reductions
$R_1,\ldots,R_s$ that can be applied to members of $\mathcal{C}_{S'}$,
with inverse expansions $R_1^{-1},\ldots, R_s^{-1}$.  Consider the
following conditions on these reductions and expansions:
\begin{itemize}

\item[(R1)] For every $N\in \mathcal{C}_{S',m}$ with $|S'|\geq 2$,
there exists some reduction $R_i$ that can be applied to $N$.

\item[(R2)] For every $N\in \mathcal{C}_{S',m}$ and for every
reduction $R_i$, $R_i(N)\in \mathcal{C}_{S'_i,m_i}$ for some
$S'_i\subseteq S'$ and $m_i< m$; moreover, $S'_i$ and $m_i$ only
depend on $S'$, $m$ and $R_i$, not on $N$.

\item[(R3)] For every $N\in \mathcal{C}_{S',m}$ and for every
reduction $R_i$, if $R_i$ can be applied to $N$, then $R_i^{-1}$ can
be applied to $R_i(N)$ and $R_i^{-1}(R_i(N))\cong N$.

\item[(R4)] For every reduction $R_i$ and for every $N,N'\in
\mathcal{C}_{S'_i,m_i}$ such that $N\cong N'$, if the corresponding
expansion $R_i^{-1}$ can be applied to $N$, then it can also be
applied to $N'$ and the resulting networks are isomorphic.
\end{itemize}

The definitions and results given in Section \ref{sec:red-1n} imply
that:
\begin{itemize}

\item The set of all $R$ and $G$ reductions satisfy conditions (R1) to
(R4) for the classes $\mathcal{C}_{S'}$ of all binary 1-nested hybridization
networks on a set $S'$.

\item The set of all $R$, $T$, and $G$ reductions satisfy conditions
(R1) to (R4) for the classes $\mathcal{C}_{S'}$ of all semibinary 1-nested
hybridization networks on a set $S'$.

\item The set of all $R$, $T$, $G$, and $\overline{G}$ reductions satisfy
conditions (R1) to (R4) for the classes $\mathcal{C}_{S'}$ of all 1-nested
hybridization networks on a set $S'$.
\end{itemize}

Now, we have the following result.

\begin{lemma}\label{lem:basic-met}
Let $S$ be a given set of labels.  For every $S'\subseteq S$, let
$\mathcal{C}_{S',m}$ and $R_1,\ldots,R_s$ be as above, and assume that
these reductions satisfy conditions (R1) to (R4).  For every
$S'\subseteq S$, let $F_{S'}:\mathcal{C}_{S'}\to X_{S'}$ be a
representation in a metric space $(X_{S'},d^{(S')})$.

Then, $F_S$ is injective up to isomorphism if the following two
conditions are satisfied for every $S'\subseteq S$, for every $m\geq
|S'|$, for every reduction $R_i$, and for every $N,N'\in
\mathcal{C}_{S',m}$ such that $F_{S'}(N)=F_{S'}(N')$:
\begin{itemize}

\item[(A)] If $R_i$ can be applied to $N$, then it can also be applied
to $N'$.

\item[(R)] If $R_i$ is applied to $N$ and $N'$, then
$F_{S'_i}(R_i(N))=F_{S'_i}(R_i(N'))$.
\end{itemize}

In particular, if these two conditions are satisfied, then
$d^{(S)}_{F_S}$ is a metric on $\mathcal{C}_S$.
\end{lemma}

\begin{proof}
We shall prove by induction on $|S'|+m$ the following statement:
\begin{quote}
For every $S'\subseteq S$ and $m\geq |S'|$, if $N,N'\in
\mathcal{C}_{S',m}$ are such that $F_{S'}(N)= F_{S'}(N')$, then
$N\cong N'$.
\end{quote}

The starting case, when $|S'|+m=2$, is obvious because then $S'$ must
be a singleton, and there is, up to isomorphism, only one 1-nested
hybridization network on a given singleton $\{i\}$: a single node
labeled with $i$.

Let now $N,N'\in \mathcal{C}_{S',m}$ be such that
$F_{S'}(N)=F_{S'}(N')$ and $|S'|+m\geq 3$.  If $|S'|=1$, we reason as
in the starting case to deduce that $N\cong N'$, so we assume that
$|S'|\geq 2$.  By (R1), some reduction $R_i$ can be applied to $N$,
and since $F_{S'}(N)=F_{S'}(N')$, by (A) it can also be applied to
$N'$.  Then, by (R2), $R_i(N), R_i(N')\in \mathcal{C}_{S'_i,m_i}$,
with $S'_i\subseteq S'$ and $m_i<m$, and by (R) we have that
$F_{S'_i}(R_i(N))=F_{S'_i}(R_i(N'))$.  Therefore the induction
hypothesis applies, implying that $R_i(N)\cong R_i(N')$.  But then, by
(R3), $R_i^{-1}$ can be applied to $R_i(N)$ and $R_i(N')$ and
$R_i^{-1}(R_i(N))\cong N$ and $R_i^{-1}(R_i(N'))\cong N'$, while, by
(R4), $R_i^{-1}(R_i(N))\cong R_i^{-1}(R_i(N'))$.  This implies that
$N\cong N'$, as we wanted to prove.

Thus, in particular, we have that for every $m\geq |S|$, if $N,N'\in
\mathcal{C}_{S,m}$ are such that $F_{S}(N)= F_{S}(N')$, then $N\cong
N'$.  Now notice that if $N,N'\in \mathcal{C}_S$, then there exists
some $m$ such that $N,N'\in \mathcal{C}_{S,m}$: take as $m$ the
largest number of nodes in $N$ or in $N'$.  Therefore, $F_S$ is
injective up to isomorphism, as we claimed.
\end{proof}

\begin{remark}
If one wants to use a result like the last lemma to prove the
injectivity up to isomorphism of a certain representation of $S$-rDAGs
more general than 1-nested networks, then it may be necessary to
explicitly add to (A) and (R) a third condition that covers the
starting case:
\begin{itemize}
\item[(S)] For every $i\in S$, $F_{\{i\}}$ is injective up to
isomorphism.
\end{itemize}
\end{remark}

We shall also use a couple of times the following straightforward
fact.

\begin{lemma}\label{lem:refines}
Let $F:\mathcal{C}\to X$ and $F':\mathcal{C}\to X'$ be two representations of $\mathcal{C}$ in
metric spaces $(X,d)$ and $(X',d')$, and assume that $F(A)=F(B)$
implies $F'(A)=F'(B)$.  Then, if $F'$ is injective up to isomorphism,
so is $F$.  \qed
\end{lemma}

When the hypothesis of this lemma is satisfied, we say that $F$
\emph{refines} $F'$, and also that $d_F$ \emph{refines} $d'_{F'}$.
Notice that if $d'_{F'}$ is a metric and $d_F$ refines it, then it is
also a metric.

\section{Robinson-Foulds distance}
\label{sec:RF}

Let $N=(V,E)$ be a $S$-rDAG. For every node $v\in V$, the
\emph{cluster} of $v$ in $N$ is the set $C(v)\subseteq S$ of leaves
that are descendants of $v$.  The \emph{cluster representation} of $N$
is the multiset
$$
\mathcal{C}(N)=\{C_N(v)\mid v\in V\},
$$
where each member appears with multiplicity the number of nodes having
it as cluster.  In particular, the cardinal of $\mathcal{C}(N)$ (as a
multiset, that is, every element counted with its multiplicity) is
equal to the number of nodes in $N$.

The \emph{Robinson-Foulds distance} between a pair of $S$-rDAGs $N,N'$
is
$$
d_{RF}(N,N')=|\mathcal{C}(N)\bigtriangleup \mathcal{C}(N')|,
$$
where the symmetric difference and its cardinal refer to multisets.
It is the natural generalization to $S$-rDAGs of the well known
Robinson-Foulds distance for phylogenetic trees
\cite{robinson.foulds:1981}.

\begin{remark}\label{rem:gtnoreg}
If $v$ is an ancestor of $u$ in $N$, then $C(u)\subseteq C(v)$, but
the converse implication is false, even in binary galled trees.  See,
for instance, Fig.~\ref{fig:rf3} below: in both networks, the root and
its tree child have the same cluster, but the root is not a descendant
of its child.
\end{remark}

It is known that the Robinson-Foulds distance is a metric on the class
of all \emph{regular} evolutionary networks on a given set $S$ (the
networks $N$ such that the mapping $v\mapsto C(v)$ induces an
isomorphism of directed graphs between $N$ and the Hasse diagram of
$(\mathcal{C}(N),\supseteq)$) \cite{baroni.ea:ac04} and on the class
of all tree-child phylogenetic networks on a given set $S$ that do not
contain any hybrid node with two parents connected by a path
\cite{cardona.ea:07a}.  Unfortunately, 1-nested networks, or even
binary galled trees, need not be regular (by Remark \ref{rem:gtnoreg})
and they can contain reticulation cycles where one merge path is a
single arc.  So, we cannot use those results to prove that the
Robinson-Foulds distance is a metric, even on the class of all
binary galled trees.

As a matter of fact, the cluster representation is not injective up to
isomorphism, and hence the Robinson-Foulds distance is not a metric,
for 1-nested networks, or even galled trees, unless we restrict the
possible in- and out-degrees of their nodes: they cannot contain
either internal tree nodes of out-degree other than 2 (see
Fig.~\ref{fig:rf1}), or hybrid nodes of out-degree 0 (see
Fig.~\ref{fig:rf4}) or greater than 1 (see Fig.~\ref{fig:rf3}).
Therefore, the Robinson-Foulds distance can only be a metric for
\emph{binary} 1-nested networks, that is, for binary galled trees.
Now, we have the following result.

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[thick,>=stealth,scale=0.25]
\draw (0,0) node[tre] (1) {}; \etq 1
\draw (2,0) node[tre] (2) {}; \etq 2
\draw (4,0) node[tre] (3) {}; \etq 3
\draw (6,0) node[tre] (4) {}; \etq 4
\draw (4,2) node[hyb] (A) {};
\draw (2,4) node[tre] (b) {};
\draw (2,7) node[tre] (a) {};
\draw (4,8) node[tre] (r) {};
\draw[->](r)--(a);
\draw [->](r)--(4);
\draw [->](r)--(A);
\draw [->](a)--(1);
\draw [->](a)--(b);
\draw [->](b)--(2);
\draw [->](b)--(A);
\draw [->](A)--(3);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[thick,>=stealth,scale=0.25]
\draw (0,0) node[tre] (1) {}; \etq 1
\draw (2,0) node[tre] (2) {}; \etq 2
\draw (4,0) node[tre] (3) {}; \etq 3
\draw (6,0) node[tre] (4) {}; \etq 4
\draw (4,2) node[hyb] (A) {};
\draw (2,4) node[tre] (b) {};
\draw (2,7) node[tre] (a) {};
\draw (4,8) node[tre] (r) {};
\draw[->](r)--(a);
\draw [->](r)--(4);
\draw [->](a)--(A);
\draw [->](a)--(1);
\draw [->](a)--(b);
\draw [->](b)--(2);
\draw [->](b)--(A);
\draw [->](A)--(3);
\end{tikzpicture}
\end{center}
\caption{\label{fig:rf1} Two non-isomorphic galled trees with the same cluster representation and internal tree nodes of out-degree 3.}
\end{figure}

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[thick,>=stealth,scale=0.3]
\draw (0,0) node[hyb] (1) {}; \etq 1
\draw (4,0) node[tre] (2) {}; \etq 2
\draw (2,2) node[tre] (a) {};
\draw (2,4)  node[tre] (r) {};
\draw[->](r)--(a);
\draw [->](r)--(1);
\draw [->](a)--(1);
\draw [->](a)--(2);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[thick,>=stealth,scale=0.3]
\draw (0,0) node[tre] (1) {}; \etq 1
\draw (4,0) node[hyb] (2) {}; \etq 2
\draw (2,2) node[tre] (a) {};
\draw (2,4)  node[tre] (r) {};
\draw[->](r)--(a);
\draw [->](r)--(2);
\draw [->](a)--(1);
\draw [->](a)--(2);
\end{tikzpicture}
\end{center}
\caption{\label{fig:rf4} Two non-isomorphic galled trees with the same cluster representation and hybrid nodes of out-degree 0.}
\end{figure}

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[thick,>=stealth,scale=0.3]
\draw (0,0) node[tre] (1) {}; \etq 1
\draw (2,0) node[tre] (2) {}; \etq 2
\draw (4,0) node[tre] (3) {}; \etq 3
\draw (6,0) node[tre] (4) {}; \etq 4
\draw (3,2) node[tre] (b) {};
\draw (0,4) node[hyb] (A) {};
\draw (6,6) node[tre] (a) {};
\draw (2,8) node[tre] (r) {};
\draw[->](r)--(a);
\draw [->](r)--(A);
\draw [->](a)--(4);
\draw [->](a)--(A);
\draw [->](A)--(1);
\draw [->](A)--(b);
\draw [->](b)--(2);
\draw [->](b)--(3);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[thick,>=stealth,scale=0.3]
\draw (0,0) node[tre] (4) {}; \etq 4
\draw (2,0) node[tre] (1) {}; \etq 1
\draw (4,0) node[tre] (2) {}; \etq 2
\draw (6,0) node[tre] (3) {}; \etq 3
\draw (5,2) node[hyb] (A) {};
\draw (3,4) node[tre] (b) {};
\draw (3,6) node[tre] (a) {};
\draw (3,8) node[tre] (r) {};
\draw[->](r)--(a);
\draw [->](r)--(A);
\draw [->](a)--(4);
\draw [->](a)--(b);
\draw [->](b)--(1);
\draw [->](b)--(A);
\draw [->](A)--(2);
\draw [->](A)--(3);
\end{tikzpicture}
\end{center}
\caption{\label{fig:rf3} Two non-isomorphic galled trees with the same cluster representation and hybrid nodes of out-degree 2.}
\end{figure}

\begin{theorem}
\label{prop:RF-AR}
Let $N,N'$ be two binary 1-nested networks on a given set $S$ such
that $\mathcal{C}(N)=\mathcal{C}(N')$.
\begin{itemize}

\item[(A)] If a specific $R$ or $G$ reduction can be applied to $N$,
then it can also be applied to $N'$.

\item[(R)] If a specific $R$ or $G$ reduction is applied to $N$ and
$N'$, the resulting networks have the same cluster representations.
\qed
\end{itemize}
\end{theorem}

In order not to lose the thread of the paper, we postpone the proof of
this theorem until \S A2 in the Appendix at the end of the paper.
Combining Lemma \ref{lem:basic-met} with this theorem, we obtain the
following result.

\begin{corollary}\label{th:RF}
The Robinson-Foulds distance is a metric on the class of all binary
galled trees on a given set $S$.  \qed
\end{corollary}

\section{Tripartitions distance}
\label{sec:trip}

Let $N=(V,E)$ be a $S$-rDAG. For every node $v\in V$, let
$A(v)\subseteq S$ be the set of (labels of) strict descendant leaves
of $v$ and $B(v)= C(v)\setminus A(v)$ the set of non-strict descendant
leaves of $v$; $B(v)$ may be empty, but $A(v)\neq\emptyset$ by Lemma
\ref{1n no ih}.  The \emph{tripartition} associated to $v$
\cite{moret.ea:2004} is
$$
\theta(v)=(A(v),B(v),S\setminus C(v)).
$$
Notice that the tripartition associated to a node $v$ refines its
cluster $C(v)$, by splitting it into $A(v)$ and $B(v)$.

The \emph{tripartitions representation} of $N$ is the multiset
$$
\theta(N)=\{\theta(v)\mid v\in V\}
$$
of tripartitions of the nodes of $N$.  The \emph{tripartitions
distance} between a pair of $S$-rDAGs $N,N'$ is
$$
d_{tri}(N,N')=|\theta(N)\bigtriangleup \theta(N')|,
$$
where the symmetric difference and its cardinal refer to multisets.

It turns out that the tripartitions distance is a metric on the class
of all 1-nested networks on a given set.  It is a consequence of the
following proposition, whose proof we postpone until \S A3 in the
Appendix.

\begin{theorem}
\label{prop:tri-AR}
Let $N,N'$ be two 1-nested networks on a given set $S$ such that
$\theta(N)= \theta(N')$.
\begin{itemize}

\item[(A)] If a specific $R$, $T$, $G$, or $\overline{G}$ reduction can be
applied to $N$, then it can also be applied to $N'$.

\item[(R)] If a specific $R$, $T$, $G$, or $\overline{G}$ reduction is applied
to $N$ and $N'$, the resulting networks have the same tripartitions
representations.  \qed
\end{itemize}
\end{theorem}

So, using Lemma \ref{lem:basic-met}, we deduce the following result.

\begin{corollary}\label{th:tri}
The tripartitions distance is a metric on the class of all 1-nested
networks on a given set $S$.  \qed
\end{corollary}

\begin{remark}\label{rem:mu}
Another refinement (in the sense of Lemma \ref{lem:refines}) of the
Robinson-Foulds distance, the so-called \emph{$\mu$-distance}, was
introduced by Cardona \textsl{et al} \cite{cardona.ea:07b} and proved
to be a metric on the class of all tree-child $S$-rDAGs for any given
$S$: then, in particular, it is a metric on the class of all 1-nested
networks on a set $S$.  Soon later, L. Nakhleh \cite{nakhleh:2009}
proposed a distance $m$ that turned out to refine the
$\mu$-distance \cite{cardona.ea:09-nak} and therefore that is also a
metric on the class of all 1-nested networks on a set $S$.  The
interested reader can look up the aforementioned references for the
specific definitions of these metrics.
\end{remark}

\section{Nodal and splitted nodal distances}
\label{subsec:nodal}

Let $N=(V,E)$ be a $S$-rDAG; to simplify the language, throughout this
section we assume that $S=\{1,\ldots,n\}$ with $n=|S|$.  Recall from
\cite{cardona.ea:tcbb:comparison.1:2008} that the \emph{least common
semistrict ancestor}, LCSA for short, of a pair of nodes $u,v\in V$ is
the node that is a common ancestor of $u$ and $v$ and strict ancestor
of at least one of them, and that is a descendant of all other nodes
in $N$ satisfying these properties.  Such a LCSA of a pair of nodes
$u,v$ always exists and it is unique \cite[\S
IV]{cardona.ea:tcbb:comparison.1:2008}, and we shall denote it by
$[u,v]$.

The LCSA of a pair of nodes in a phylogenetic tree is their lowest
common ancestor.  It turns out that such a characterization extends to
1-nested networks.  Recall that a \emph{lowest common ancestor}, LCA
for short, of a pair of nodes $u,v$ in a rDAG is any common ancestor
of $u$ and $v$ that is not a proper ancestor of any other common
ancestor of them \cite{bender.ea:2005}.

\begin{lemma}\label{lem:lcsa-1n}
Every pair of nodes $u,v$ in a 1-nested network has only one LCA, and
it is their LCSA.
\end{lemma}

\begin{proof}
Let $x$ be any LCA of $u$ and $v$, and let us prove that $x$ must be a
strict ancestor of $u$ or $v$.  Indeed, by Lemma \ref{lem:RF-previ} in
\S A1 in the Appendix, if $x$ is not a strict ancestor of $u$, then it
is intermediate in the reticulation cycle for a hybrid node $h_u$ that
is a strict ancestor of $u$.  In a similar way, if $x$ is not a strict
ancestor of $v$, then it is intermediate in the reticulation cycle for
a hybrid node $h_v$ that is a strict ancestor of $v$.  Now, if $x$
were not a strict ancestor either of $u$ or of $v$, then it either
would happen that it is intermediate in reticulation cycles for two
different hybrid nodes, which is impossible in a 1-nested network, or
that it is a proper ancestor of a common ancestor of $u$ and $v$,
namely $h_u=h_v$, against the assumption that $x$ is a LCA of $u$ and
$v$.

So, $x$ is a common ancestor of $u$ and $v$ and a strict ancestor of
at least one of them, and thus it is an ancestor of $[u,v]$.  Since
$x$ cannot have proper descendants that are common ancestors of $u$
and $v$, we conclude that $x=[u,v]$.
\end{proof}

For every pair of leaves $i,j\in S$, let $\ell_N(i,j)$ and
$\ell_N(j,i)$ be the distances from $[i,j]$ to $i$ and to $j$,
respectively, and let $\nu_N(i,j)=\ell_N(i,j) +\ell_N(j,i)$.

The \emph{LCSA-path lengths matrix} of $N$ is the symmetric matrix
$$
\nu(N)=\left(\begin{array}{ccc} \nu_N(1,1) & \ldots & \nu_N(1,n)\\
\vdots & \ddots & \vdots\\
\nu_N(n,1) & \ldots & \nu_N(n,n)
\end{array}
\right)
$$
and the \emph{splitted LCSA-path lengths matrix} of $N$ is the (not
necessarily symmetric) matrix
$$
\ell(N)=\left(\begin{array}{ccc} \ell_N(1,1) & \ldots & \ell_N(1,n)\\
\vdots & \ddots & \vdots\\
\ell_N(n,1) & \ldots & \ell_N(n,n)
\end{array}
\right)
$$

The \emph{nodal distance} between a pair of $S$-rDAGs $N,N'$ is half
the Manhattan, or $L_1$, distance between $\nu(N)$ and $\nu(N')$:
$$
d_{\nu}(N,N')=\frac{1}{2} \sum_{1\leq i\neq j\leq n}
|\nu_N(i,j)-\nu_{N'}(i,j)|.
$$
The \emph{splitted nodal distance} between $N$ and $N'$ is the
Manhattan distance between $\ell(N)$ and $\ell(N')$:
$$
d_{\ell}(N,N')=\sum_{1\leq i\neq j\leq n}
|\ell_N(i,j)-\ell_{N'}(i,j)|.
$$
Of course, instead of using the Manhattan distance on the set of
$n\times n$ matrices, one can use any other distance for real-valued
matrices to compare LCSA-path lengths, or splitted LCSA-path lengths,
matrices, like for instance the euclidean distance.  The results in
this section do not depend on the actual metric for real-valued
matrices used.

The nodal distance $d_{\nu}$ is the natural generalization to
$S$-rDAGs of the classical nodal metric for binary phylogenetic trees
\cite{farris:sz73,willcliff:taxon71}, while the splitted nodal
distance $d_{\ell}$ generalizes to $S$-rDAGs the recently introduced
homonymous metric for arbitrary phylogenetic trees
\cite{cardona.ea:jmb:2008}.

It is known \cite{nodalnetworks,cardona.ea:tcbb:comparison.2:2008}
that $d_{\nu}$ is a metric on the class of all binary tree-child time
consistent phylogenetic networks on a given set $S$, and that $d_\ell$
is a metric on the class of all tree-child time consistent
phylogenetic networks on a given set $S$, but no binary galled tree
containing a reticulation cycle with one merge path consisting of a
single arc is time consistent, and therefore we cannot use these
results to prove that $d_{\nu}$ or $d_\ell$ are metrics even for
binary galled trees.

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[thick,>=stealth,scale=0.3]
\draw (0,0) node[tre] (1) {}; \etq 1
\draw (3,2) node[hyb] (A) {};
\draw (3,0) node[tre] (2) {}; \etq 2
\draw (1,3) node[tre] (a) {};
\draw (2,5)  node[tre] (r) {};
\draw[->](r)--(a);
\draw [->](r)--(A);
\draw [->](a)--(1);
\draw [->](a)--(A);
\draw [->](A)--(2);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[thick,>=stealth,scale=0.3]
\draw (0,0) node[tre] (2) {}; \etq 2
\draw (3,2) node[hyb] (A) {}; \etq 2
\draw (3,0) node[tre] (1) {}; \etq 1
\draw (1,3) node[tre] (a) {};
\draw (2,5)  node[tre] (r) {};
\draw[->](r)--(a);
\draw [->](r)--(A);
\draw [->](a)--(2);
\draw [->](a)--(A);
\draw [->](A)--(1);
\end{tikzpicture}
\end{center}
\caption{\label{fig:nod0} Two non-isomorphic binary galled trees on $S=\{1,2\}$ and whith the same LCSA-path length, 3, between their only two leaves.}
\end{figure}

It turns out that $\nu$ is not injective up to isomorphism, and hence
$d_{\nu}$ is not a metric, even for binary galled trees, as
Fig.~\ref{fig:nod0} shows.  As far as $\ell$ goes, it is not injective
up to isomorphism for 1-nested networks, or even galled trees, that
are not semibinary: if we allow hybrid nodes of out-degree 0 (see
Fig.~\ref{fig:nod1}) or greater than 1 (see Fig.~\ref{fig:nod2}),
there exist pairs of non-isomorphic galled trees with the same
splitted LCSA-path length matrices.  Therefore, $d_\ell$ can be a
metric at most on the class of all semibinary 1-nested networks.  Now,
we have the following result.

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[thick,>=stealth,scale=0.3]
\draw (0,0) node[tre] (1) {}; \etq 1
\draw (4,0) node[tre] (2) {}; \etq 2
\draw (2,4)  node[tre] (r) {};
\draw[->](r)--(1);
\draw [->](r)--(2);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[thick,>=stealth,scale=0.3]
\draw (0,0) node[tre] (1) {}; \etq 1
\draw (4,0) node[hyb] (2) {}; \etq 2
\draw (1,2) node[tre] (a) {};
\draw (2,4)  node[tre] (r) {};
\draw[->](r)--(a);
\draw [->](r)--(2);
\draw [->](a)--(1);
\draw [->](a)--(2);
\end{tikzpicture}
\end{center}
\caption{\label{fig:nod1} Two non-isomorphic galled trees with the same $\ell$ matrix and hybrid nodes of out-degree 0.}
\end{figure}

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[thick,>=stealth,scale=0.3]
\draw (0,0) node[tre] (1) {}; \etq 1
\draw (2,0) node[tre] (2) {}; \etq 2
\draw (4,0) node[tre] (3) {}; \etq 3
\draw (3,2)  node[tre] (a) {};
\draw (2,6)  node[tre] (r) {};
\draw[->](r)--(1);
\draw [->](r)--(a);
\draw [->](a)--(2);
\draw [->](a)--(3);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[thick,>=stealth,scale=0.3]
\draw (0,0) node[tre] (1) {}; \etq 1
\draw (2,0) node[tre] (2) {}; \etq 2
\draw (4,0) node[tre] (3) {}; \etq 3
\draw (3,2) node[hyb] (A) {};
\draw (1,4) node[tre] (a) {};
\draw (2,6)  node[tre] (r) {};
\draw[->](r)--(a);
\draw [->](r)--(A);
\draw [->](a)--(1);
\draw [->](a)--(A);
\draw [->](A)--(2);
\draw [->](A)--(3);
\end{tikzpicture}
\end{center}
\caption{\label{fig:nod2} Two non-isomorphic galled trees with the same $\ell$ matrix and hybrid nodes of out-degree greater than 1.}
\end{figure}

\begin{theorem}
\label{prop:snod-AR}
Let $N,N'$ be two semibinary 1-nested networks on a given set $S$ such
that $\ell(N)= \ell(N')$.
\begin{itemize}

\item[(A)] If a specific $R$, $T$, or $G$ reduction can be applied to
$N$, then it can also be applied to $N'$.

\item[(R)] If a specific $R$, $T$, $G$ reduction is applied to $N$ and
$N'$, the resulting networks have the same splitted LCSA-path lengths
matrices.  \qed
\end{itemize}
\end{theorem}

As we did previously, we postpone the proof of this theorem until \S
A4 in the Appendix at the end of the paper.  Combining Lemma
\ref{lem:basic-met} with this theorem, we obtain the following result.

\begin{corollary}\label{th:nod}
The splitted nodal distance is a metric on the class of all semibinary
1-nested networks on a given set $S$.  \qed
\end{corollary}

\section{Conclusion}

Several slightly different definitions of galled tree, capturing the
notion of a hybridization network with isolated reticulation cycles,
have been proposed so far in the literature.  The most general such
definition is as a network with arc-disjoint reticulation cycles
\cite{CJSS2005,jansson.sung:2006}, called in this paper
\emph{1-nested}, and the most restrictive is Gusfield \textsl{et al}'s
original definition of a galled tree as a network with node-disjoint
reticulation cycles \cite{gusfield.ea:csb:2003}: in between lie the
level-1 networks of Janson, Sung \textsl{et al}
\cite{JanssonSung2004a,JanssonSung2008}.  In the \emph{semibinary}
(hybrid nodes of in-degree 2 and out-degree 1) case, level-1 and
1-nested networks are the same, and in the \emph{binary} (semibinary
plus tree nodes of out-degree 2) case, galled trees, level-1 networks
and 1-nested networks are the same objects.

In this paper we have established for which classes of 1-nested
networks on a fixed set of labels, several distance measures
introduced so far in the literature satisfy the axioms of metrics:
actually, only the separation axiom (distance 0 means isomorphism) is
relevant here, because all other axioms of metrics are always
satisfied by these distances.  In summary, we have proved that:

\begin{enumerate}[(a)]
\item The Robinson-Foulds distance
\cite{baroni.ea:ac04,cardona.ea:tcbb:comparison.1:2008} is a metric
only for binary galled trees.

\item The tripartitions distance \cite{moret.ea:2004}, the
$\mu$-distance \cite{cardona.ea:07b} and Nakhleh's metric $m$ for
reduced networks \cite{nakhleh:2009} are metrics for arbitrary
1-nested networks.

\item The natural translation of the nodal distance for phylogenetic
trees to evolutionary networks \cite{nodalnetworks} is not a metric
even for binary galled trees.

\item The splitted nodal distance
\cite{nodalnetworks,cardona.ea:tcbb:comparison.2:2008} is a metric for
semibinary 1-nested networks, but not for
arbitrary galled trees.
\end{enumerate}
We would like to mention that the 1-nested networks turn out to
form the first well-defined class of evolutionary networks where
the tripartitions distance is shown to be a metric but the
Robinson-Foulds distance is not a metric.

There are other distances that have not been discussed in this paper
because they obviously fail to be metrics even for binary galled
trees.  This is the case of the triplets distance
\cite{cardona.ea:tcbb:comparison.2:2008}, which cannot be a metric for
binary galled trees because there are many more binary galled trees
with 3 leaves than possible triplets in the sense defined in the
aforementioned paper.  And, as it was already observed in \cite[\S
II.D]{cardona.ea:07b}, it is also the case of any distance defined by
comparing the multisets of induced subtrees, or the multisets of
splits of induced subtrees: for instance, the pairs of galled trees
depicted in Figs.  \ref{fig:nod1} or \ref{fig:nod2} have the same
multisets of induced subtrees.

The splitted nodal distance and the triplets distance were introduced
in \cite{cardona.ea:tcbb:comparison.2:2008} as suitable
generalizations of the corresponding distances for phylogenetic
networks with the aim of obtaining metrics on the class of tree-child
time consistent phylogenetic networks, and hence they were not
designed to cope with reticulation cycles where one merge path is a
single arc.  This is the main reason of their failure as metrics for
arbitrary 1-nested networks.  But it seems not difficult to modify
them to obtain metrics for 1-nested networks, by taking into
account the restricted, and specific, topological structure of these
networks: something similar was already done with the splitted nodal
distance to make it work on tree-child time consistent evolutionary
networks with hybrid nodes of (almost) arbitrary type
\cite{nodalnetworks}.

Galled trees, 1-nested networks, and level-1 networks are defined as having hybrid nodes of in-degree 2, in the first case by semantical reasons and in the other two cases for practical reasons (to guarantee that certain reconstruction algorithms run in polynomial time), and we have kept this restriction in this paper. But, although Gusfield \textsl{et al}'s node-disjoint reticulation cycles condition implies that hybrid nodes must have in-degree 2,  this restriction is not necessary in level-1 and 1-nested networks, and polynomial time algorithms for the reconstruction of level-1 or 1-nested networks with hybrid nodes of arbitrary in-degree may be discovered in the future, in which case it would be interesting to know whether the distance measures discussed in this paper define metrics in this more general case and they can be used thus to assess these new algorithms.

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text:
Gustav Theodor Drobisch (1811-1882), écrivain et publiciste allemand ;
 Moritz Wilhelm Drobisch (1802-1896), philosophe, logicien et mathématicien allemand ;
 Till Drobisch (né en 1993), coureur cycliste namibien.

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text:
Member of Parliament for Wantage
The Member of Parliament for Wantage - Check out David's maiden speech below. 👇🏻👇🏻
The Wantage constituency covers part of the Vale of White Horse and part of South Oxfordshire.
The largest town in the constituency is Didcot, which grew up around the railway when Brunel put his London to Bristol line through Didcot instead of Abingdon.
There are three market towns in the constituency – Wantage, Faringdon and Wallingford. Wantage is famous as the birthplace of King Alfred, while Wallingford has had a Royal Charter since 1155. Faringdon still bears the scars of the English civil war, when its church lost the top of its steeple.
The constituency is overwhelmingly rural in character, with more than 400 farms still in operation. It is marked by the famous White Horse and the ancient Ridgeway runs along its southern border.
The constituency is also one of the most economically dynamic places in the country. Harwell, the birthplace of nuclear power in the UK, now houses some of the most important scientific establishments in the world, including the Diamond Synchrotron, ISIS and ISIS 2. There are numerous world-class businesses based in the constituency, including Williams Formula One, Oxford Instruments and RM plc.
With close links to London, Oxford, Reading and Swindon, there are many commuters who live in the constituency but work elsewhere
District Councillors
Police and Crime Commissioner Candidate
Help us take action on local issues and build a better Britain.
We rely on the support of individuals like you.


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text:
Category: SM Use
Social Media Handbook for Police: Part 11
Welcome to the the next instalment in my series of social media tips. These are aimed primarily at a police audience, but hopefully applicable to a wider group of people too, especially those in the public sector. This series of posts will aim to identify some good practice and useful hints and tips for police officers and staff to consider when using social media.
Part 11: More Operational Uses
Part 10 of the handbook tackled using social media in public order. As I said there, one of the more common themes around police use of social media is the question of how it can be used operationally. There is often a lot of scepticism – it is fine for 'engagement' but not for so called 'real policing'. A number of forward thinking forces and individuals have however made a great deal of progress using social media in the more operational areas of policing.
Social media has been used in investigation of both serious and less serious crimes. Many people will be aware of the Joanna Yates case involving Avon and Somerset police. There is still a facebook page up in memory of Joanna Yates, and the criminal justice system needs to be aware of the potential for public discussion of cases and the impact it may have on investigation, and ultimately on the ability to hold a fair trial. As an aside the Facebook page also contains some unpleasant comments, including several from people professing to be the killer.
Recently a DCI from Norfolk police was quoted as saying that "social media sites can be a doubl- edged sword. In major investigations they can find they have to deal with a large amount of unreliable information, posted online. Social media sites increase the volume of information police have to sift through, sometimes making it harder to identify what really happened." This was in relation to a murder case, and centred around difficulties identifying people with real first hand knowledge as opposed to people who were just re-tweeting rumours and second hand information.
This can make social media a challenging space for investigation; the sense of anonymity and ease of spreading stories and rumours that social media can generate can swiftly result in a torrent of information that makes managing an investigation difficult. However it is always possible that there is real intelligence in all the other posts, and possibly even a genuine confession.
I recently advised on whether to use social media to make a witness appeal for a high profile case – my advice was to be very clear about what outcomes were expected, and be careful about allowing people to post up on a Facebook page or similar. I also suggested that the demographic of likely witnesses should be considered, and adequate resources put in place to sift through the replies as they come in. In many ways the process is similar to that which an SIO goes through when considering any press release or media briefing.
My final piece of advice was to use a pre-existing account where possible, that people already had trust and confidence in – creating a new account will usually only work for the most high profile cases. If you want significant numbers of people to see and act on a social media request for witnesses, then using an account that already has a reasonable number of followers is the way forward. Of course, this means that you need to start now, and not wait until the major incident occurs…
This post was previously published on Partrdigej's blog.
Using Twitter Hashtags for Emergency Management by Scott Mills
Seizing the Virtual Scene by Lauri Stevens
West Midlands (UK) Police: Twitter on the Frontline by Mark Payne
Previous posts from the Social Media Handbook Series:
Part 1: What Social Media networks should I use?
Part 2: How do I get followers / friends ???
Part 3: Policies / Strategies / Guidance??
Part 4: Ten things to have on your page to drive up interest??
Part 5: What to do when things go wrong
Part 6: We don't do that here
Part 7: Basic Guides – Twitter and Flick'r
Part 8: Connect it all together
Part 9: Talk to local people
Part 10: Operational Uses
Justin Partridge
Justin Partridge is a senior manager for Lincolnshire Police in England. He also works on Local Policing and Partnerships for the Association of Chief Police Officers (ACPO).
Justin Partridge has worked in the public sector since leaving university, and for the police since 2003. After being one of only three non-sworn staff selected for the prestigious Police Strategic Command Course (for those who aspire to the most senior posts in UK policing), he started working on the national Local Policing and Partnerships area with chief officers from across the UK, and with partners from the Home Office, NPIA, APA and elsewhere.
Justin is passionate about making a difference to people, and see social media and new technologies having a major role in this – especially in policing and the wider public sector. He blogs on a variety of issues, predominantly around police and technology, and can be found on Twitter talking about much the same.
Social Media Quick Tip: Clean Up Your Twitter Following
If you follow a lot more people than follow you back, Twitter won't allow you to follow more people
There are tens of thousands of third-party applications to help us use Twitter. Hundreds of those are designed specifically to help determine who to unfollow. Why would you want to unfollow someone? Two reasons to unfollow someone are:
1. Because the account is inactive; and
2. Because they're not following you.
It's good to keep your following well-maintained. If you follow a lot more people than follow you back, Twitter will not allow you to follow more people. Additionally, in case it matters to you, on Twitter, it's not considered good form. Although there are many tools to help with this, here are two I use often:
1. Untweeps.com: This handy little tool allows you to sign in with your Twitter account and see the people you follow who haven't tweeted in months. You can select how many days it's been and then select who to unfollow.
2. JustUnfollow.com: Sign in with your Twitter credentials. The next screen allows you to decide if you want to view unfollowers (those you follow who don't follow you) or fans (those who follow you who you don't follow). Both are good to take a look at and decide for yourself if you'd like to make adjustments.
This Social Media Quicktip was previously published on LawOfficer.com.
QuickTips, SM Use, Twitter
Portland OR Police Bureau embracing social media
In May 2011, I had the opportunity to attend the SMILE Conference in Chicago and I came back incredibly energized with moving forward on our use of social media. One of the ideas presented at the conference was from Chief Grogan of Dunwoody, GA, about promoting an event via social media to draw followers.
On Friday June 3, 2011, the Portland Police Bureau released the artwork for its new police car logos via Facebook. A news release was sent out 2 days prior and informed the community that the new design would be launched on Facebook. We encouraged people to come "like" us to see the design.
Portland Police Bureau Facebook numbers had been growing at roughly 40 per week but in the 3 days leading up to and following the vehicle design launch, 451 new people "liked" the Portland Police Bureau. Twitter followers jumped as well and the hope is that it will continue to increase because the Portland Police Bureau's official Twitter handle (@PortlandPolice) is going to be on the new patrol cars. In addition, the Bureau also maintains a second Twitter handle (@ppbpio) for media releases and information.
In the very near future, the goal is to have officers in each of the three precincts and the Traffic Division maintaining Twitter and Facebook accounts to connect directly with the communities they serve. The Bureau has recently purchased iPads for the two PIOs and is looking at smart phones for officers to use in the field.
The Portland Police Bureau is developing a strategic communications plan and social media is playing a big role in its development. The Office of Public Information is staffed by two full-time sworn Public Information Officers, and three non-sworn members. Part of the communications plan includes a push toward social media journalism and producing stories to be delivered directly to the community via social media platforms.
Though traditional media remains a constant news stream for community members, the ability to tell our own story directly to community members via social media is beneficial. Recent studies conclude that people are getting their news increasingly from social media and our agency is working toward being connected with those people.
Become a "friend" of the Portland Police Bureau at Facebook.com/PortlandPolice and follow us on Twitter @PortlandPolice.
Sergeant Pete Simpson is an 18-year-veteran of the Portland Police Bureau. Sergeant Simpson worked in the Gang Enforcement Team for 11 years as an officer and detective before promoting to Sergeant in 2008. Sergeant Simpson was a Hostage Negotiator for 7 years as well as an instructor for the National Gang Center in Tallahassee, Florida. Sergeant Simpson is currently assigned to the Chief's Office as one of the Public Information Officer's for the Portland Police Bureau.
June, 30
Case Studies, Facebook, SM Use, SMILE, Twitter
Using Social Media for an Investigative and Tactical Advantage
Law enforcement officers in general are reluctant to change, especially when it comes to new technology. For that reason many departments and their people are behind the "electronic eight ball."
Social media has interconnected our lives more than we think and best of all it is free. You can't miss an advertisement on television or in print that doesn't end with the "Follow Us" tag. It is the biggest thing for law enforcement since the use of DNA in criminal investigations. For law enforcement, social media provides a treasure trove of information without many of the traditional legal hurdles involved to obtain it. But are you or your department embracing this developing technology? I remember when the job turned from typewriters to computers and there was a mass exodus of the "old timers" and with them left a tremendous amount of experience. Social media is here to stay and might just make your investigative life easier.
Law enforcement finds itself at the electronic crossroads where intelligence gathering and investigations meet. Intelligence is important, but real-time intelligence is invaluable. Investigators now have the ability of developing real-time intelligence with just a couple of clicks or swipes. Everyone these days is a handheld journalist. It might not be someone posting directly at the scene but a witness from another vantage point chattering about it. The question remains, "Are you listening?" Investigative teams and especially supervisors when planning the traditional Canvass for Additional Witnesses, the Surveillance Camera Canvass, etc., should have an investigator devoted entirely to conducting a Social Media Canvass. One properly trained and versed investigator armed with a department issued tablet or smartphone can accomplish this task. However there is a caveat. As law enforcement officers we always want to use the secret passage into the "other world" and not the general admission door. You never want to use your personal account that can be traced back to you and your family. Departments must address this situation.
A suspect's "conversations" aren't the only ones that need to be monitored during the Social Media Canvass. Quite often the traditional news posts information from the suspect's social media pages. If you look at the way departments use social media it almost seems as if they are apprehensive. Social media for law enforcement is a give and take situation. The department disseminates information to the general public, but should also want to collect information. I examined hundreds of police department social media sites and found that many don't follow their own local media pages and I have to ask the question, "Why not?" Law enforcement agencies should be plugged into local media sites as well as any other page that provides useful information.
In addition to the investigative resource, law enforcement must embrace the new social media revolution as a tactical one as well. For instance, during a hostage crisis in Utah on June 21, 2011, the hostage taker used social media to communicate with his family. However, his friends that were on the scene posted messages to him about SWAT deployments, which could have made a bad situation worse. Hostage negotiators often look to get the conversation started and interview family and friends for help. Negotiators should ask during the interviews if the person has a penchant for social media and if they do what user names are involved.
Social media is transforming the way we conduct investigations and should be treated like every other major change we have gone through. Social media cannot be an individual choice of whether or not it should be used, but embraced by the department as a whole. When investigators glean productive and actionable information from social media, they should protect it as operational security – keeping secrets secret. If not, we may see similar situations that make our job more difficult such as suspect's bleaching crime scenes, setting fires and the use of throw away phones.
Joseph L Giacalone
Joseph L. Giacalone is a 19 year NYPD Detective Sergeant with an extensive background in criminal investigations. He has held many prestigious positions, but his favorite was the Commanding Officer of the Cold Case Homicide Squad. Joe obtained a Master of Arts Degree in Criminal Justice with a Specialty in Crime and Deviance from John Jay College of Criminal Justice in 2005. He has been an Adjunct Professor at John Jay since January of 2006 and is the author of the Criminal Investigative Function: A Guide for New Investigators, published by Looseleaf Law. You can follow Joe on Twitter: @ColdCaseSquad or @JoeGiacalone or on the web at: coldcasesquad.com
5 Tips for Mixing Social Media and the Press
Social Media offers good opportunities for your department to get information out to the public and tell it's own story without the help of the media. But sometimes the story could use a bit more audience reach in a short period of time to achieve your goal. This means getting your story to the press. Here are some tips for getting the press to cover your stories when you need them to using both social media and a bit of your own personal charm.
1. Make Friends.
Get to know the reporters covering police topics in your area. Meet with them in person and develop a good rapport. If they know you and have a friendly relationship with you, they are more likely to help you in a pinch. Make sure to follow and be-friend local reporters, media outlets and assignment desks on Twitter and Facebook. Sometimes a media outlet needs a last minute story and they may post and ask for an idea or check on Twitter or Facebook for interesting news to cover.
2. Go Old School When You Need To.
Although it's been a growing trend for police departments at least in my area to communicate with reporters mostly via email regarding details of an event, keep in mind that a phone call can go a long way towards getting your story covered correctly and favorably. This is mostly because it's easier for a reporter to ask a stream of questions, get answers directly back, and clarify details with you in a live conversation.
3. Give 'Em a Shout.
If a reporter or media outlet gives you a hand with a story, give them a shout out and thank them on Facebook and Twitter. On the other hand, if a reporter or media outlet posts a negative story, skip the urge to call them out in public. Mistakes or rogue journalism is much better handled professionally and in private. Remember, what's posted on the Internet stays out there a loooooooooong time.
4. 10-7 at 101.
When posting about topics on Twitter or Facebook use a conversational tone. Use of police jargon is easy to fall into, but you'll get much more response from the public and the media regarding a story if you use plain old English. Not to mention, you'll get fewer phone calls from the press asking you to clarify what you meant.
5. Time Is Not Always On Your Side.
Release your information in a timely manner. If you want the press to cover a story, even if it's good news, get that information out there via a tweet, a post, an emailed press release, or phone call to your local reporter. Old news is not news at all and the older a story is, the less chance the media will cover it.
Media, SM Use
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Home / In the Media / Border Police `lie about violence at fence protests'
Border Police `lie about violence at fence protests'
in In the Media July 29, 2005
By Jonathan Lis
Ha'aretz Daily
For more than six months, dozens of Israelis and hundreds of Palestinians have been demonstrating every weekend against the construction of the separation fence near the West Bank village of Bil'in. These demonstrations, defined by participants as peaceful, frequently turn into violent clashes with the Border Police's Company 22, assigned to disperse the demonstrations.
An investigation by Haaretz has found that policemen from that company have made false accusations against demonstrators and even made arrests on the basis of those accusations. Palestinians thus detained can be held for eight days before being brought before a judge.
In other cases, soldiers gave false testimony about rock throwing and other violence when most of the protests were peaceful. However, there were instances in which Border Police were hurt by rocks thrown by demonstrators at Bil'in.
In recent weeks, three judges harshly criticized troops after watching videotapes that nullified their allegations. In at least two cases the judges questioned the excessive force used against peaceful and restrained protesters. Footage taken by Shai Caremeli-Polack, who documents the demonstrations for the organization Anarchists Against the Wall, was presented in the various court hearings and contradicted the claims of Border policemen that the conduct of demonstrators had compelled them to use batons, kick and fire tear gas directly at the demonstrators.
In the wake of one of these incidents, police officials this week harshly criticized the behavior of Company 22.
Two of the cases examined by the courts or the police's Internal Affairs Department (IAD) are detailed here. Last Friday, Bil'in residents demonstrated with Israeli and overseas human rights activists. Border Policemen arrested a Beit Lakia resident (name withheld). The troops transferred the suspect to the Judea and Samaria Police for questioning, on the grounds that he had assaulted a policeman.
Anarchists Against the Wall activists rushed to the station and showed investigators their complete documentation of the incident. The videotape, which Haaretz has also obtained, clearly shows the Border policemen violently attacking the man and kicking him after they had subdued him and held him prone on the ground. "After the investigator saw the film, he immediately decided to release the detainee and transfered the case file to IAD to review the Border Policemen's conduct," a police officer said yesterday.
On July 20, Border Police arrested two Israeli demonstrators at Bil'in – Shaul Berger Mugrabi and Moshe Robas, claiming they had participated in an illegal assembly, assaulted a policeman and interfered with police work. Jerusalem Magistrate's Court Judge Yoel Tzur watched the videotape of the incident and lambasted the police: "The Border policemen who were involved in the incident are indeed framing the two respondents, yet I cannot shake off the impression created by the tape that shows distinctly that it was Border Policemen who used force against the respondents."
The Border Police responded: "The protesters against the fence construction knowingly break the law in entering a closed military zone, with their objective being to thwart the construction of the security fence. These cases are not about the right to demonstrate and freedom of expression, since the demonstrations in question are not authorized and illegal."
According to the Border Police, "in contrast to the claims of protesters that these are peaceful demonstrations, for two and a half years, almost daily, IDF soldiers and Border Police fighters have been contending with demonstrations characterized by serious violence, hurled rocks and Molotov cocktails that endanger the fighters' lives. In recent months, IDF soldiers and Border Police fighters have been severely and moderately wounded during demonstrations in Bil'in […] The videotapes handed to the reporter are one-sided and do not reflect the complete picture and the violent conduct of the demonstrators. The police has tapes which in most of the cases proves precisely the opposite."
Bil'in	2005-07-29
Tagged with: Bil'in
Imprisoned Human Rights Defender in Military Court
Than you for the donations to free Ashraf Abu Rahmah!
12th anniversary of the popular resistance in Bil'in