text:
\section{Introduction}
	\label{sec:Introduction}

	In this paper we study the neighborhood polynomial of graphs and give an algorithm to compute the polynomial for chordal graphs in polynomial time for some subclasses.
	Throughout the paper, all graphs are simple, finite and undirected.
	For a graph $G = (V,E)$, the \emph{neighborhood} of a vertex $v \in V$ is the set of all adjacent vertices, denoted by $N_G(v) = \{u \in V \mid uv \in E \}$.
	The \emph{neighborhood complex} of a graph $G$, first introduced by \cite{Lovasz1978}, consists of all subsets of vertices $W \subseteq V$ which have a common neighbor, that is
	\[
		 \mathcal{N}_G = \{U \subseteq V | \: \exists v \in V: U \subseteq N_G(v) \}.
	\]
	This set-system is clearly hereditary and hence it is a simplicial complex.
	To count the number of sets with cardinality $k$ in $\mathcal{N}_G$, we define the \emph{neighborhood polynomial}
	\[
		N_G(x) = \sum_{U \in \;\mathcal{N}_G} x^{|U|},
	\]
	which is the generating function of the neighborhood complex $\mathcal{N}_G$.
	Since we only consider finite graphs, the sum is finite and $N_G(x)$ is a polynomial such as all other generating functions considered in this paper.
	We investigate the complexity of computing the neighborhood polynomial of some graph classes. In particular, we look at chordal graphs and subclasses like interval graphs, split graphs and chordal comparability graphs.
	In order to do this, we introduce the \emph{anchor width} of a graph and develop an algorithm for computing the neighborhood polynomial in Section~\ref{sec:anchorDegree}. We will see that the anchor width is the essential parameter for a polynomial runtime of our algorithm. If for any subclass of chordal graphs the anchor width is polynomially bounded in the number of vertices, our algorithm is efficient. In particular our main result is the following theorem.
	\begin{theorem} \label{thm:runtime}
		Let $G$ be a chordal graph with $n$ vertices and anchor width $k$. Computing the neighborhood polynomial takes at most $\mathcal{O}(n^3 k + n^2k^2)$ time.
	\end{theorem}
	In Section~\ref{sec:AnchorWidth} we investigate the complexity of the anchor width for different subclasses.
	For this we look at chordal graphs with bounded leafage. The leafage $\ell(G)$ was introduced in~\cite{LinMcKeeWest1998} and measures how close a chordal graph is to an interval graph.
	We show that a chordal graph~$G$ on $n$ vertices has anchor width at most $n^{\ell(G)}$ (cf. Theorem~\ref{thm:leafage}). Furthermore for interval graphs, which are the graphs with leafage at most two, we give a family with quadratic anchor width.
	Another result of Section~\ref{sec:AnchorWidth} is that chordal comparability graphs have linearly bounded anchor width.

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

	The neighborhood polynomial was introduced by \cite{BrownNowakowski2008} who investigated the effect of some elementary graph operations on the neighborhood polynomial. Given two graphs $G_1=(V_1,E_1)$ and $G_2 = (V_2,E_2)$ on disjoint vertex sets, the \emph{union} $G_1 \cup G_2$ of the graphs is the graph on the vertex set $V_1 \cup V_2$ with edge set $E_1 \cup E_2$.
	The \emph{join} $G_1 +G_2$ of the two graphs is the graph on the vertex set $V_1 \cup V_2$ consisting of both graphs together with all possible edges between vertices in $V_1$ and vertices in $V_2$, that is $E = E_1 \cup E_2 \cup \{ v_1v_2 \mid  v_1 \in V_1, v_2 \in V_2\}$.

	\begin{proposition}[\cite{BrownNowakowski2008}]
		\label{prop:DisjointUnion}
		Let $G_1$ and $G_2$ be two graphs on disjoint vertex sets.
		Then the neighborhood polynomial of the disjoint union $G_1 \cup G_2$ is
		\[
		N_{G_1 \cup G_2}(x) = N_{G_1}(x) + N_{G_2}(x) -1 .
		\]
	\end{proposition}

	\begin{proposition}[\cite{BrownNowakowski2008}]
		\label{prop:Join}
		Let $G_1 = (V_1,E_1), G_2 = (V_2,E_2)$ be two graphs on disjoint vertex sets.
		Then the neighborhood polynomial of the join $G_1 + G_2$ is
		\[
		N_{G_1+ G_2}(x) = (1+x)^{|V_2|}N_{G_1}(x) + (1+x)^{|V_1|}N_{G_2}(x)- N_{G_1}(x )N_{G_2}(x).
		\]
	\end{proposition}

	The two graph operations, disjoint union and join, are used to define cographs. \emph{Cographs} are exactly the graphs which do not contain an induced $P_4$, a path on four vertices. They can be constructed recursively. Starting with a single vertex as a cograph, the disjoint union and the join of two cographs are cographs.
	For this and other well-known graph theoretic facts, we refer to~\cite{Golumbic1980}.
	The neighborhood polynomial of a single vertex graph is $N(K_1,x) =1$ and the two operations disjoint union and join, given by the two formulas in Proposition~\ref{prop:DisjointUnion} and Proposition~\ref{prop:Join} are computable in linear time. Note that $(1+x)^n$ can be computed in linear time using the binomial theorem. \cite{CorneilPerlStewart1985} present a linear time algorithm to recognize cographs and give the corresponding recursive construction rules using disjoint union and join. Hence the neighborhood polynomial of a cograph is computable in quadratic time.

	Another graph operation is attaching one vertex $v$ to a subset of vertices of a graph~$G$. This operation was studied by \cite{AlipourTittmann2021}, who gave an explicit formula for a neighborhood polynomial after attaching a vertex to a subset of vertices. More formally for a graph $G = (V,E)$, a subset $U \subseteq V$ of vertices and an additional vertex $v \notin V$, we denote by $G_{U \triangleright v}$ the graph with vertex set $V \cup \{v\}$ and edge set $E \cup \{uv \mid u \in U\}$.
	For simplification we use the following notation for all $W\subseteq V$:
	\begin{align*}
		N_G^{\cap}(W) &= \bigcap_{w \in W} N_G(w) \quad \text{and} \\
		N_G^{\cup}(W) &= \bigcup_{w \in W} N_G(w).
	\end{align*}

	\begin{proposition}[\cite{AlipourTittmann2021}]
		\label{prop:AlipourTittmannUpdate}
		Let $G = (V,E)$ be a graph, $U \subseteq V$ and $v \notin V$.
		Then the neighborhood polynomial of $G_{U \triangleright v}$ is
		\[
		N_{G_{U \triangleright v}}(x) = N_G(x) + \sum\limits_{\substack{W \subseteq U, \\W \neq \emptyset}} \phi_W + \sum\limits_{\substack{W \subseteq U, \\ W \neq \emptyset}} (-1)^{|W|+1} x(1+x)^{|N_G^{\cap}(W)|},
		\]
		where
		\[
		\phi_W =
		\begin{cases}
		x^{|W|},&\text{ if } N_G^{\cap}(W) = \emptyset; \\
		0,&\text{ otherwise.}
		\end{cases}
		\]
	\end{proposition}

	Using this formula, Alipour and Tittmann showed that for a fixed integer~$k$, computing the neighborhood polynomial of $k$-degenerate graphs is possible in polynomial time, see~\cite{AlipourTittmann2021}.
	A \emph{$k$-degenerate graph} is a graph where every subgraph has a vertex~$v$ with $\deg(v) \leq k$.
	Using the degeneracy, we can pick one vertex of degree $\leq k$ after another and update the neighborhood polynomial by the formula of Proposition \ref{prop:AlipourTittmannUpdate} in order to get a polynomial runtime.
	As a corollary it follows that there is a polynomial-time algorithm to compute the neighborhood polynomial for planar (or more general graphs of bounded genus) and $k$-regular graphs, see \cite{AlipourTittmann2021}.
	This update formula of Alipour and Tittman (see Proposition~\ref{prop:AlipourTittmannUpdate}) was the starting point of our investigations for chordal graphs.

	A graph $G$ is said to be \emph{chordal} if there is no induced cycle of length $\geq 4$.
	Equivalently a graph is chordal if and only if it has a perfect elimination order.
	A \emph{perfect elimination order} is an ordering $v_1, \ldots, v_n$ of the vertices such that for all $i$ the neighborhood of $v_i$ in $G[\{v_i, \ldots v_n \}]$ is a clique.
	Here for a subset $U \subseteq V$ the graph $G[U]$ denotes the subgraph of $G$ induced by $U$.
	A vertex, whose neighborhood is a clique is called \emph{simplicial}.
	It is well-known that every non-trivial chordal graph has at least two simplicial vertices, which gives us the perfect elimination order (cf. \cite{Golumbic1980}).
	In order to study the neighborhood polynomial of chordal graphs and their subclasses, we make use of the perfect elimination order to build the chordal graph by attaching one vertex after another to a clique.
	We adapt the formula of Alipour and Tittmann (Proposition~\ref{prop:AlipourTittmannUpdate}) to our use.
	To get some complexity results of computing the neighborhood polynomial, the connection to the domination polynomial is useful. For this we introduce dominating sets.
	A \emph{dominating set} of a graph $G = (V,E)$ is a set of vertices $D \subseteq V$ such that every vertex of the graph is either in $D$ or adjacent to a vertex of $D$, i.e.
	\begin{align*}
	D \ \cup N^{\cup}_G(D) = V.
	\end{align*}
	The family of all dominating sets of a graph $G$ is denoted by $\mathcal{D}_G$ and the \emph{domination polynomial} $D_G(x)$ is the generating function of $\mathcal{D}_G$ that is
	\begin{align*}
	D_G(x) = \sum_{U \in \mathcal{D}_G} x^{|U|}.
	\end{align*}
	The following relation between domination polynomials and neighborhood polynomials holds. For a proof see for example \cite{HeinrichTittmann2018}.
	\begin{proposition} \label{prop:connectionDominating}
		For a graph $G = (V,E)$ and its complement graph $\overline{G}$ it holds:
		\begin{align*}
		D_{\overline{G}}(x) + N_G(x) = (1+x)^{|V|}.
		\end{align*}
	\end{proposition}
	With other words this proposition states that every vertex set has either a common neighbor in the graph or is a dominating set in the complement graph.
	The connection of these two polynomials can be used to determine the complexity of computing the neighborhood polynomial. In particular, the neighborhood polynomial is computable in polynomial time if and only if the domination polynomial of the complement graph is computable in polynomial time.
	Furthermore the contributions to the well-known graph problem DOMSET, imply some complexity results for the neighborhood polynomial. DOMSET is the problem of
	deciding whether a graph has a dominating set of size $\leq k$ for a given $k$.
	\begin{corollary} \label{col:domsethard}
		Let $\mathcal{G}$ be a class of graphs and $\overline{\mathcal{G}}$ the class of the complement graphs of $\mathcal{G}$.
		If DOMSET is \NP-complete on $\overline{\mathcal{G}}$, then computing the neighborhood polynomial on $\mathcal{G}$ is \NP-hard.
	\end{corollary}

	DOMSET is \NP-hard on
	many graph classes such as chordal graphs, see~\cite{BoothJohnson1982}.
	\cite{Bertossi1984} showed that it is \NP-hard on bipartite graphs and split graphs. \emph{Split graphs} are the graphs where the vertex set can be partitioned into a clique and an independent set. Since split graphs are exactly the graphs which are chordal and \emph{co-chordal} (i.e. the complement graph is chordal), DOMSET is also \NP-hard on co-chordal graphs.
	This together with Corollary~\ref{col:domsethard} shows the \NP-hardness of computing the neighborhood polynomial in split graphs (cf.~\cite{Day2017}) and hence in chordal graphs.

	\section{Algorithm for Chordal Graphs}
	\label{sec:anchorDegree}

	Our algorithm relies on the perfect elimination order of chordal graphs and comes from the vertex-attachment formula of Alipour and Tittmann, see Proposition~\ref{prop:AlipourTittmannUpdate}.
	First, we adapt their formula to our special case where we attach a vertex to a clique.
	To study the new arising neighborhood sets after vertex attachment, we introduce anchor sets, which are subsets of a clique appearing as a common neighborhood of a set of vertices.
	The maximal number of anchor sets of a clique, which we denote as anchor width, is the essential parameter in our algorithm to get a polynomial runtime. \\

	\begin{figure}[tb]
		\centering
		\includegraphics[scale = 0.6,page = 1]{peripherie}
		\caption{An illustration of the introduced sets, the periphery $P_C$ and a periphery set $M$ with corresponding anchor set $A_G(M,C)$ of a clique $C$. }
		\label{fig:illustrationDefinition}
	\end{figure}

	Let $C$ be a clique in a graph $G = (V,E)$.
	We define the set of neighbors of the clique $C$, not including the clique itself as the \emph{periphery} of $C$, denoted by
	\begin{align*}
	P_G(C) = N^{\cup}_G(C) \backslash C.
	\end{align*}
	A subset $M \subseteq P_G(C)$ of the periphery is called \emph{periphery set}. Note that the empty set is also a periphery set.
	We call a non-empty subset $A$ of $C$ \emph{anchor set}, if it is the common neighborhood in $C$ of some periphery set $M$. See Figure~\ref{fig:illustrationDefinition} for an illustration. In general not all subsets of a clique are an anchor set.
	For every $M \subseteq P_G(C)$ we define the \emph{corresponding anchor set} in $C$ as
	\begin{align*}
	A_G(M,C) =  N^{\cap}_G(M) \cap C
	\end{align*}
	if the intersection is non-empty. The neighborhood of the empty periphery set $M = \emptyset$ is the empty intersection and thus the corresponding anchor set is $C$.
	Moreover, several periphery sets $M$ and $M'$ can correspond to the same anchor set $A_G(M,C) = A_G(M',C)$.
	For an anchor set $A$ of $C$, the periphery sets $M \subseteq P_G(C)$ whose common  neighborhood in $C$ is $A$ build the family
	\[
	\mathcal{P}_G(A,C) = \left\{M \subseteq P_G(C) \mid A_G(M,C) = A \right\}.
	\]
	The generating function of $\mathcal{P}_G(A,C)$ is called \emph{periphery polynomial} and defined by
	\[
	P_G(A,C,x) = \sum_{M \in \mathcal{P}_G(A,C)} x^{|M|}.
	\]
	Note that $\mathcal{P}_G(A,C) = \emptyset$ and $P_G(A,C,x) =0$ if $A$ is not an anchor set of $C$.
	The family of all anchor sets of a clique $C$ is
	\[
	\mathcal{A}_G(C) = \{A \subseteq C \mid  A \neq \emptyset \text{ and } \exists M \subseteq P_G(C):  A = A_G(M,C)\}.
	\]
	Note that $C \in  \mathcal{A}_G(C)$ for every clique $C$, since $C$ is the anchor set of the empty periphery set.
	The \emph{anchor width} of a graph $G$ is the smallest number $k$ such that $|\mathcal{A}_G(C) | \leq k$ for all cliques $C$ in $G$.

	For a maximal clique $C_{max}$ and a clique $C$ contained in $C_{max}$ the following relations hold.

	\begin{lemma}\label{lem:InclusionClique}
		Let $C$ be a clique and $C_{max}$ a maximal clique containing $C$ in a graph $G= (V,E)$.
		Then the following conditions hold:
		\begin{enumerate}[(a)]
			\item $C_{max}\backslash C \subseteq P_G(C) \subseteq P_G(C_{max}) \cup (C_{max} \backslash C)$
			\item $\mathcal{A}_G(C) = \{A \cap C \mid A \cap C \neq \emptyset \text{ and }A \in \mathcal{A}_G(C_{max})\} $
			\item For every $A \in \mathcal{A}_G(C)$ the periphery polynomial is
			\[
			P_G(A,C,x)  = (1+x)^{|C_{max}\backslash C|}\sum_{ \substack{A' \in \mathcal{A}_G(C_{max})\\ A' \cap C = A}}  P_G(A',C_{max},x).
			\]
		\end{enumerate}
	\end{lemma}

\begin{proof}
	\textit{(a)} Since every vertex in the clique $C_{max} \backslash C$ is adjacent to~$C$, the first inclusion holds.
	Furthermore, every element which is adjacent to one of the elements in $C$ is either an element of $C_{max} \backslash C$ or it is adjacent to an element of~$C_{max}$.

	To show \textit{(b)}, we check which subsets of the clique $C$ can appear as an anchor set. Let $M' \subseteq P_G(C)$ be a periphery set such that $A_G(M,C)$ is a non-empty anchor set. Using \textit{(a)} we distinguish three cases.

	If $M' \subseteq C_{max} \backslash C$, the anchor set of $M'$ is $C$ itself.
	If $M' \subseteq P_G(C_{max})$ there exists an anchor set $A = A_G(M,C_{max})$. Since $M'$ only consists of elements of the periphery of $C$, the intersection of $A$ with $C$ provides the anchor set $A_G(M,C)$, which is non-empty.
	In the final case, $M'$ consist of elements of $C_{max} \backslash C$ and $P_G(C_{max})$. We only need to consider $M = M' \cap P_G(C_{max})$, since the elements of $C_{max} \backslash C$ just lead to another intersection with $C$. We continue as in the second case.

	On the other hand, a set $A' = A \cap C \neq \emptyset $ for $ A \in \mathcal{A}_G(C_{max})$ is always an anchor set.
	For $A$ there exists a periphery set $M \subseteq P_G(C_{max})$ which has $A$ as common neighborhood $A_G(M,C_{max})$ in $C_{max}$. If we take all elements of $M$ which are in the periphery of $C$, the common intersection of those elements inside $C$ is exactly $A'$. Hence $A'$ is an anchor set.

	\textit{(c)} The periphery polynomial $P_G(A,C,x)$ counts the different periphery sets with respective to the size, where $A$ is the corresponding anchor set.
	As we have seen in \textit{(b)}, an anchor set $A$ is given by $A = A' \cap C$ for an anchor set $A' \in \mathcal{A}_G(C_{max})$. Since there are different possibilities to choose $A'$, we sum over all corresponding periphery polynomials which are counted in $P_G(A',C,x)$.
	Furthermore all elements in $C_{max }\backslash C$ are in the periphery of $C$ (see (a)) with neighborhood $C$.
	Hence we can add elements of $C_{max} \backslash C$ to any periphery set $M'$ with anchor set $A_G(M',C_{max}) = A'$ and still have as anchor set $A_G(M,C) = A$.
	For the polynomial as generating function, we multiply $P_G(A',C,x)$ by \mbox{$(1+x)^{|C_{max}\backslash C|}$}.

	For different anchor sets $A'$ and $A''$ with $A'' \cap C = A = A' \cap C$, the corresponding periphery sets are pairwise different, since the common neighborhood inside $C_{max}$ is different.
	So in order to get all periphery sets corresponding to $A$, we need to sum up these the polynomials.
\end{proof}

	This shows that it is sufficient to provide the information about anchor sets and periphery polynomials for all maximal cliques of the graph. With this information we are able to compute the necessary information for all other cliques. Furthermore the anchor width only depends on the size of the anchor family of the maximal cliques.
	\medskip

	In the following, we derive a formula for the neighborhood polynomial after vertex attachment using the periphery polynomial.
	For every set $U \subseteq V$ of vertices we define the \emph{local neighborhood} $\mathcal{N}_G(U)$ of $U$ as the family
	consisting of all vertex sets of $G$ which have a common neighbor in~$U$, that is
	\[
	\mathcal{N}_G(U) = \{ W \subseteq V \mid \exists v \in U: \ W \subseteq N_G(v) \}.
	\]
	Note that the local neighborhood $\mathcal{N}_G(V)$ of the whole vertex set is equal to the neighborhood complex $\mathcal{N}_G$.
	For every clique $C$, we can partition the local neighborhood $\mathcal{N}_{G}(C)$ by the following lemma into the disjoint sets
	\begin{align*}
	\mathcal{N}_G(A,C) = \{N \in \mathcal{N}_G(C) \mid  N \cap P_G(C) \in \mathcal{P}_{G}(A,C) \}, \quad A \in \mathcal{A}_G(C).
	\end{align*}

	\begin{lemma}\label{lem:generatingfunctNcG1}
		For every clique $C$ of the graph $G$, it holds
		\[
		\mathcal{N}_G(C) = \mathop{\dot \bigcup}_{A \in \mathcal{A}_G(C)} \mathcal{N}_G(A,C).
		\]
	\end{lemma}
	\begin{proof}
		For every $N \in \mathcal{N}_G(C)$ there is by definition a clique-vertex $v \in C$ which is adjacent to every element in $N$. Hence the common neighborhood of $N \cap P_G(C)$ inside $C$ is non-empty. Note that in general $N$ is not a periphery set since it can contain elements from $C$. The common neighborhood is an anchor set $A$. Since these anchor sets differ for different families $\mathcal{N}_G(A,C)$ the union is disjoint.
	\end{proof}

	Lemma \ref{lem:generatingfunctNcG1} is useful since we only have to determine the generating functions of $\mathcal{N}_G(A,C)$ for every $A \in \mathcal{A}_G(C)$. Adding these generating functions, we maintain the generating function of the local neighborhood $\mathcal{N}_G(C)$.
	In the next lemma, we derive a formula to compute the generating function of
	$\mathcal{N}_{G}(A,C)$  for every $A \in \mathcal{A}_G(C)$.

	\begin{lemma}\label{lem:generatingfunctNcG2}
		For a given anchor set $A \in \mathcal{A}_G(C)$ of a clique $C$, the generating function of $\mathcal{N}_{G}(A,C)$ is
		\[
		N_G(A,C,x) = P_G(A,C,x) \left( (1+x)^{|C|}- x^{|A|}(1+x)^{|C|-|A|} \right).
		\]
	\end{lemma}
	\begin{proof}
		We count the number of sets with respect to the cardinality in $\mathcal{N}_{G}(A,C)$.
		Every $M \in \mathcal{P}_G(A,C)$ is in $\mathcal{N}_{G}(A,C)$. Hence $P_G(A,C,x)$ must be a summand of $N_G(A,C,x)$.
		Furthermore there are supersets $N$ for all $M$ which contribute to $N_G(A,C,x)$.
		Since we look at all $M \in \mathcal{P}_G(A,C)$, it is enough to look at supersets $N = M \cup X$, where $X$ is a subset of $C$.
		In order to keep $N$ in the local neighborhood $\mathcal{N}_G(C)$, we need a common neighbor in $C$. Since the common neighborhood of $M$ inside $C$ is the anchor set $A$, the common neighborhood of $N$ must contain an element of $A$. Hence $X$ cannot be the whole anchor set $A$.
		In particular, the possibilities to extend $M$ are the elements of the family
		\[
		\mathcal{X} = \{X \mid \exists a \in A: X \subseteq C \backslash \{a\}  \}.
		\]
		All sets in $\mathcal{X}$ consist of a disjoint union of a proper subset of $A$ and a subset of $C \backslash A$. This leads to the generating function
		\[
		\left( (1+x)^{|A|} - x^{|A|} \right)(1+x)^{|C|-|A|}
		= (1+x)^{|C|}- x^{|A|}(1+x)^{|C|-|A|}
		\]
		of $\mathcal{X}$.
		The generating function of $\mathcal{P}_G(A,C)$, which counts the different possibilities of $M$ is counted by $P_G(A,C,x)$.
	\end{proof}

	We are now ready to prove the update formula for the neighborhood polynomial after attaching a vertex to a clique. The proof and the formula are similar to Proposition~\ref{prop:AlipourTittmannUpdate} (cf.~\cite{AlipourTittmann2021}).
	\begin{proposition} \label{cor:AttachmentClique}
		Let $G = (V,E)$ be a graph and $C$ a clique in the graph.
		The neighborhood polynomial of $G_{C \triangleright v}$ with vertex set $V \cup \{v\}$ is:
		\begin{align*
		N_{G_{C \triangleright v}}(x) &= N_G(x)+\phi_G(C) \\
		&
		+ \ x{\sum\limits_{A \in \mathcal{A}_G(C)}}{P_G(A,C,x) \left( (1+x)^{|C|}- x^{|A|}(1+x)^{|C|-|A|} \right)},
		\end{align*}
		where
		\[
		\phi_G(C) = \begin{cases}
		x^{|C|}, &\text{ if } C \text{ is a maximal clique in } G;\\
		0, &\text{ otherwise }.
		\end{cases}
		\]
	\end{proposition}

	\begin{proof}
		Let $X \in \mathcal{N}_{G_{C \triangleright v}}$ be a neighborhood set of the graph $G_{C \triangleright v}$. We consider the following three cases:
		\begin{itemize}
			\item If $X \subseteq V$ and $X \not \subseteq C$, then $X\in \mathcal{N}_G$ is in the neighborhood complex of $G$. Hence $X$ is considered in the first summand $N_G(x)$ of the above formula.
			\item Now let $X \subseteq V$ and $X \subseteq C$. If $X$ is a proper subset of $C$, it already has a common neighbor in $G$, hence it is already counted in the first summand.
			Similarly this holds if $X = C$ and $C$ is not a maximal clique in $G$.

			Thus the only case where a new neighborhood arises is if $X = C$ and $C$ is a maximal clique in $G$.
			In $G_{C \triangleright v}$ the common neighbor of $C$ is $v$. This is counted in the summand in $\phi_G(C)$.
			\item Let us now consider the case $v \in X$, i.e. $ X \not \subseteq V$.
			Since $v$ is connected to all elements in $C$, we need to count all subsets $Y \subseteq V$ which have a common neighbor in $C$. This is equivalent to count the number of elements in $\mathcal{N}_G(C)$. Combining Lemma \ref{lem:generatingfunctNcG1} and Lemma \ref{lem:generatingfunctNcG2}, we obtain
			\[
			\sum_{A \in \mathcal{A}_G(C)} P_G(A,C,x) \left( (1+x)^{|C|}- x^{|A|}(1+x)^{|C|-|A|} \right)
			\]
			as the generating function of $\mathcal{N}_G(C)$. In $X$ there is one additional element~$v$. Hence we have to multiply the polynomial with $x$.
		\end{itemize}
		Since the above cases are disjoint, this leads to the formula of the neighborhood polynomial as stated.
	\end{proof}

	With this formula, we are able to compute the neighborhood polynomial after attaching a vertex $v$ to a clique $C$ in a graph. We will use this in connection with the perfect elimination order of the graph to compute the neighborhood polynomial of chordal graphs. For this we start with a single vertex and add in reverse order of the perfect elimination order the vertex one after another. After every step we compute the neighborhood polynomial using Proposition~\ref{cor:AttachmentClique}. Furthermore we need to make sure to have the correct data to compute the neighborhood polynomial after every step. Hence we need to update the anchor family and periphery polynomials.

	To go into a more detailed analysis we assume that the chordal graph is connected.
	If the chordal graph is not connected we apply the algorithm for every connected component and compute the neighborhood polynomial of the whole graph by adding the neighborhood polynomials of every connected component (cf. Proposition~\ref{prop:DisjointUnion}). This operation can be done in linear time.

	For a connected chordal graph we compute the elimination order $v_1, \ldots , v_n$ using a lexicographic breadth-first search which has linear runtime.
	We then add the vertices in reverse order, starting with $v_n$. The neighborhood polynomial of this starting graph is $1$. Then we add $v_i$ to the corresponding clique in $G[v_{i+1}, \ldots, v_n]$ and compute the new neighborhood polynomial with Proposition~\ref{cor:AttachmentClique}.

	In order to compute the formula of Proposition~\ref{cor:AttachmentClique}, we need the anchor family of $C$ and the corresponding periphery polynomials $P_G(A,C,x)$ for every $A \in \mathcal{A}_G(C)$.
	With Lemma~\ref{lem:InclusionClique} it suffices to store these informations for the maximal cliques of the graph and compute them in every step for the required clique $C$ where we attach the new vertex.
	The details of updating this information will be explained in the next paragraph. Furthermore we have a list of the maximal cliques in the current graph which need to be updated, too. Since every step gives at most one new maximal clique, there are at most $n$ maximal cliques in a chordal graph with $n$ vertices.
	 After attaching one vertex, updating the polynomial and the corresponding information, we go on with the next step.  \\

	We now study how to update the anchor families and periphery polynomials for the maximal cliques after attaching a vertex in order to have the correct ones in the next step.
	Fix a clique $C$ of the graph $G$. The graph with attached vertex $v$ to $C$ is denoted by $G^+ = G_{C \triangleright  v}$.
	We get exactly one new maximal clique $C^+ = C \cup \{v\}$ which we have to add to the list of maximal cliques in the graph. If $C$ is a maximal clique in $G$ we have to delete the entry $C$ from the list of maximal cliques.

	We determine the anchor sets and periphery polynomial of the new arising maximal clique $C^+$.
	The periphery of $C^+$ in $G^+$ is $P_{G^+}(C^+) = P_G(C)$ and the family of anchor sets is $\mathcal{A}_{G^+}(C^+) = \mathcal{A}_{G}(C) \cup \{ C^+\}$ if $C$ was not maximal in $G$ and $\mathcal{A}_{G^+}(C^+) = \mathcal{A}_{G}(C) \backslash \{C\} \cup \{ C^+\}$ if $C$ was maximal in $G$. The periphery polynomials stay as in $G$ that is $P_{G^+}(A,C^+,x) = P_G(A,C,x)$ for all $A \in \mathcal{A}_{G}(C)$ and $P_{G^+}(C^+,C^+,x) = 1$.

	Now we go through the list of maximal cliques and update the necessary information if needed.
	The maximal cliques in $G$ which have no intersection with $C$, do not change in~$G^+$.
	Let $C_{max}$ be a maximal clique in $G$ with $C_{max} \cap C \neq \emptyset$. It is $C \neq C_{max}$.
	Note that we deleted $C$ from the list in the case that $C$ was maximal in $G$.
	The periphery of $C_{max}$ in $G^+$ consists of the periphery of $C_{max}$ in $G$ together with the new element $v$. More formally it holds
	\begin{align*}
	P_{G^+}(C_{max})=P_G(C)\cup \{v\}.
	\end{align*}
	Every anchor set of~$C_{max}$ in $G$ remains an anchor set in~$G^+$.
	Since the new vertex $v$ is attached to the subset $C_{max} \cap C$ of the considered clique $C_{max}$, this subset $C_{max} \cap C \neq \emptyset$ is a new anchor set in $G^+$, if it was not already an anchor set in $G$.
	Furthermore all subsets of $C_{max}$ occurring as non-empty intersection of $C_{max} \cap C$ with an anchor set in $\mathcal{A}_G(C_{max})$ build an anchor set of $C_{max}$ in~$G^+$.
	The family of anchor sets of $C_{max}$ in $G^+$ is:
	\begin{align} \label{eq:newanchor}
	&\mathcal{A}_{G^+}(C_{max}) = \\
	&\mathcal{A}_G(C_{max})

	\cup \{A \cap (C_{max} \cap C) \mid A \cap (C_{max} \cap C) \neq \emptyset \text{ and } A \in \mathcal{A}_G(C_{max}) \}. \nonumber
	\end{align}
	Note that  $C_{max} \cap C$ is an element of the second set in the above equation since $C_{max} \in \mathcal{A}_G(C_{max})$.
	To determine the periphery polynomial $P_{G^+}(A,C_{max},x)$ for every anchor set $A \in \mathcal{A}_{G^+}(C_{max})$, we need to distinguish the following three cases.
	Since $C \neq C_{max}$, the intersection $C_{max} \cap C $ is a proper subset of $C_{max}$.
	\begin{enumerate}[(i)]
		\item If $A$ is a proper subset of $C \cap C_{max}$, all corresponding periphery sets in $G$ are a corresponding periphery set in $G^+$ and we can add $v$ to every corresponding periphery set $M$ in $G$, since the intersection with the neighborhood $N_{G^+} (v) = C$ does not change the anchor set. In this case we get:
		\begin{align*}
					P_{G^+}(A,C_{max},x) = (1+x) P_G(A,C_{max},x).
		\end{align*}

		\item If $A = C \cap C_{max}$, the periphery sets in $G$ with corresponding anchor set $A$ which are counted in $P_G(A , C_{max} , x)$ still have the same anchor set in $G^+$.
		Furthermore $v$ is a new periphery set with anchor set $A = C \cap C_{max}$ and all periphery sets which have a superset $A'$ of $A$ as corresponding anchor set, form together with $v$ a periphery set with anchor set $A$.
		Hence the updated periphery polynomial is
		\begin{align*}
		&P_{G^+}(A , C_{max} , x) =
		P_G(A , C_{max} , x) + x \left(1 + \sum_{\substack{A' \supseteq A \\ A' \in \mathcal{A}_{G}(C_{max})} } P_G(A',C_{max},x) \right)
		\end{align*}
		\label{item:updateperiphery}
		\item
		In the remaining case $A$ is not a subset of $C \cap C_{max}$, hence $v$ is not in a periphery set with anchor set $A$. So the periphery polynomial stays the same, which means
		\[
		P_{G^+}(A, C_{max},x) = P_G(A,C_{max},x).
		\]
	\end{enumerate}

	This concludes the analysis of the cases and hence the algorithm to compute the neighborhood polynomial of a chordal graph.
	This algorithm leads to a polynomial time algorithm if the anchor width is polynomially bounded as stated in Theorem~\ref{thm:runtime}.
	We now give a detailed analysis of the runtime which concludes the proof of Theorem~\ref{thm:runtime}.

	\begin{proof}[of Theorem~\ref{thm:runtime}]
			Using lexicographic breadth-first search, we get a perfect elimination order of the chordal graph $G$ in linear time (cf.~\cite{Golumbic1980}).
			After attaching one vertex, there is one new maximal clique, containing~$v$. We add this maximal clique to the list of all maximal cliques in the graph, possibly removing the neighborhood of $v$ if it was in the list.
			This shows that we have at most $n$ maximal cliques.
			Furthermore we have an ordering of the vertices given by the perfect elimination order which is the reverse order of attaching the vertices to the graph.
			Hence in order to compute an intersection or test a subset relation, we only need to compare the elements in the clique in the attached order. Both is possible in linear time.

			For every step we need to update the neighborhood polynomial and afterwards the other informations. In total we attach $n$ vertices and hence have $n$ steps.

			To update the neighborhood polynomial after vertex attachment to a clique $C$, we first need to find a maximal clique, containing $C$ and compute its anchor family and its periphery polynomials.
			In order to find this maximal clique we test for every maximal clique (at most $n$) whether $C$ is a subset. This needs at most $\mathcal{O}(n^2)$ time.

			Computing the anchor family takes $\mathcal{O}(nk)$ and for one anchor set the periphery polynomial takes $\mathcal{O}(nk)$ and since we have at most $k$ anchor sets in a clique, we need $\mathcal{O}(nk^2)$ for this step (cf. Lemma~\ref{lem:InclusionClique}).
			Now using the update formula to compute the neighborhood polynomial is possible in $\mathcal{O}(k)$.
			Updating the neighborhood polynomial takes for every step at most $O(nk^2)$ time.

			Afterwards we update the anchor families and periphery polynomials for all maximal cliques with non-empty intersection with $C$.
			We have at most $n$ maximal cliques, where we need to update this information.
			Updating the anchor family for one maximal clique takes $\mathcal{O}(nk)$ time (see (\ref{eq:newanchor}))
			Hence for all maximal cliques this takes at most $\mathcal{O}(n^2k)$.
			For one anchor set, updating the periphery polynomials takes at most $\mathcal{O}(nk)$ in case~(\ref{item:updateperiphery}). This case appears at most once and compute the periphery polynomial in the other cases is possible in linear time.
			Updating the necessary information for anchor sets and their periphery polynomials leads to a runtime of at most $\mathcal{O}(n^2k)$ for all maximal cliques.
			Since we add in total $n$ vertices, one after another, this leads to a total runtime of $\mathcal{O}(n^3k + n^2k^2 )$.
	\end{proof}

	\section{Complexity of the  Anchor Width}
	\label{sec:AnchorWidth}
	In this section, we will discuss some subclasses of chordal graphs and study their anchor width. We show that there are subclasses with polynomially bounded anchor width. We arrived at these graph classes starting from interval graphs, the first class for which we found a polynomial bound.
	For these subclasses the algorithm explained in Section~\ref{sec:anchorDegree} runs in polynomial time.
	In contrast to this result, we show that the anchor width of split graphs, a simple well-known subclass, is not polynomially bounded (see the following proposition). Hence the algorithm introduced in Section \ref{sec:anchorDegree} might take super-polynomial time. \\

	\begin{proposition} \label{prop:splitexponential}
		For all $n \in \mathbb{N}$ there is a split graph on $n$ vertices with anchor width at least $2^{\frac{n}{2}}-1$.
	\end{proposition}

	\begin{proof}
		We construct an infinite family of split graphs $S_m$ on $n = 2m $ vertices such that the anchor width is $2^m-1$.
		We start with a clique $C = \{c_1, \ldots , c_m\}$ of size~$m$ and attach vertices $v_1, \ldots, v_m$  such that every $v_i$ ($1 \leq i \leq m$) is adjacent to $c_j$ for all $j \neq i$.
		This constructed graph is a split graph since $C$ is a clique and $\{v_1, \ldots, v_m \}$ an independent set.
		All vertices $v_i$ are in the periphery $P_C$, hence $|P_C| = m= \frac{n}{2}$.
		For a periphery set $M \subseteq P_C$ we denote by $I_M$ the indices of those vertices $v_i$ which are in $M$.
		The corresponding anchor set is then $\{C_j \mid j \notin I_M \}$. In such a way we can construct every non-empty subset of $C$ as an anchor set.
		Hence the anchor width of $S_m$ is $2^m -1 = 2^{\frac{n}{2}}-1$.
	\end{proof}

	Another interesting family of subclasses are the chordal graphs with bounded leafage. For those graphs we can show a polynomial upper bound of the anchor width.
	The leafage is a parameter which stems from the intersection graph representation of chordal graphs.
	An \emph{intersection graph} of a family of sets $\mathcal{F}$ is the graph consisting of one vertex for every set in $\mathcal{F}$ such that two vertices are adjacent if and only if the corresponding sets have a non-empty intersection.
	An interval graph is an intersection graph of a family of subtrees of a path and
	chordal graphs are exactly the graphs which are the intersection graph of a family of subtrees of a \emph{host tree}~\cite{Gavril1974}.
	We call a representation of a chordal graph by a family of subtrees a \emph{subtree representation}.
	\cite{LinMcKeeWest1998} introduced the leafage of a chordal graph, which measures how close a chordal graph is to an interval graph.
	More precisely, the \emph{leafage} $\ell(G)$ of a chordal graph $G$ is defined as the minimum number of leaves of the host tree among all subtree representations of $G$.
	We call a subtree representation \emph{optimal} if it has the minimum number of leaves in the host tree.
	The interval graphs are exactly the chordal graphs with leafage at most $2$.
	The split graphs $S_m$ constructed in Proposition~\ref{prop:splitexponential} have leafage $m$ and a possible optimal host tree is the star $K_{1,m}$.
	\cite{HabibStacho2009} present a polynomial-time algorithm in order to compute the leafage of a chordal graph.
	As mentioned in \cite{LinMcKeeWest1998} we may restrict to host trees whose number of vertices is the number of maximal cliques of $G$.
	\begin{lemma}
		\label{lem:1-1corresondence}
		There exists an optimal representation such that the vertices of the host tree are in one-to-one correspondence with the maximal cliques of the graph.
	\end{lemma}
	\begin{proof}
		Since every pairwise intersecting family of subtrees has the {Helly property}, i.e.\ the intersection of all subtrees of a subtree representation of a clique is non-empty~\cite{Golumbic1980}, there is at least one common vertex $v_C$ for every clique $C$ in the host tree.
		A vertex in the host tree cannot belong to different maximal cliques since their union has to form a clique as well and hence the cliques would not be maximal.
		Furthermore all subtrees intersecting in a vertex $v$ of the host tree build a clique $C$.
		If $C$ is not maximal, there is a maximal clique $C_{max}$ containing $C$. Contracting the path from $v$ to $v_{C_{max}}$ in the host tree does not increase the number of leaves.
		Thus, if we choose an optimal representation with few vertices as possible, the claim follows.
	\end{proof}

	We study the connection between the anchor width and the leafage of a chordal graph and show an upper bound of the anchor width.
	In the following, we identify a subtree of the host tree by its vertices.\\

	\begin{theorem}\label{thm:leafage}
		For a chordal graph $G$ with leafage $\ell = \ell(G)$ and $n$ vertices, the anchor width is at most $n^{\ell}$.
	\end{theorem}

	\begin{proof}
		Let $C = C_{max}$ be a maximal clique in the graph $G$. We consider an optimal subtree representation of $G$ such that the vertices of the host tree $T$ are in one-to-one correspondence with the maximal cliques of $G$ (cf. Lemma \ref{lem:1-1corresondence}).
		Let $v_C$ be the vertex in the host tree which corresponds to the clique~$C$ of $G$.
		From $v_C$ there is a unique path in the host tree to all $\ell$ leaves which we denote by $P_1, \ldots , P_{\ell}$.

		For a periphery set $M \subseteq P_C$,
		the corresponding anchor set consists of those elements of the clique whose neighborhood contains $M$.
		For every $w \in M$, there is a tree $T_w$ representing $w$ in the subtree configuration.
		Since $C$ is a maximal clique, these trees $T_w$ do not contain $v_C$ since otherwise $w$ would belong to $C$.
		For every path $P_i$, we define a vertex $v_i$ representing $M$ on $P_i$ as follows:
		\[	v_i \in \argmin_{v \in T_{w_i} \cap P_i} dist(v,v_c),\]
		where $w_i$ is an element from the periphery such that
		\[
		w_i \in \argmax_{w \in M} \min_{v \in T_w \cap P_i} dist(v,v_c).
		\]
		So for every $w \in M$ such that $T_w \cap P_i \neq \emptyset$, we choose the closest vertex $v_w$ to $v_C$ on the path $P_i$ of the corresponding tree $T_w$. Among those vertices $\{v_w\}_w$, the vertex $v_i$ is the vertex with maximal distance to $v_C$.
		If there is no subtree $T_w$ of the periphery which has a non-empty intersection with the path $P_i$, we set $v_i = v_C$.
		Note that the $v_i$'s are not necessarily distinct.

		Now the anchor set $A = A_G(M,C)$ consists exactly of all subtrees of the clique $C$, which contain all $v_1, \ldots, v_{\ell}$ and $v_C$. If there is no such subtree corresponding to an element of the clique, there is no corresponding anchor set to $M$ in $C$.
		The anchor set $A$ is fully determined by the vertices $v_i$.

		A chordal graph with $n$ vertices has at most $n$ maximal cliques. Hence the host tree has at most $n$ vertices which gives at most $n$ choices for every $v_i$.
		In total we have at most $n^{\ell}$ choices for the tuple $(v_1, \ldots, v_{\ell})$ and hence at most $n^{\ell}$ different anchor sets.
		This shows the upper bound for the anchor width.
	\end{proof}

	Since interval graphs are the graphs with leafage at most $2$, it follows:
	\begin{corollary}
		The anchor width of interval graphs on $n$ vertices is at most $n^2$.
	\end{corollary}

	\begin{figure}[tb]
			\centering
		\includegraphics[scale=0.65]{IntervalQuadraticOptimal}
		\caption{Construction of an interval graph with $21$
			vertices and a maximal clique of size $11$
			and $25$ anchor sets. }
		\label{fig:IntervalQuadraticBound}
	\end{figure}

	The magnitude of the bound is optimal for interval graphs since there is an infinite family of interval graphs on $n = 4m +1$ vertices with a clique of size $2m+1$ which has at least $m^2 = \left( \frac{(n-1)}{4}\right)^2$ different anchor sets.
	For the construction (see Figure \ref{fig:IntervalQuadraticBound}), we take the path $P$ on $2m+1$ vertices $v_{-m}, \ldots, v_0,\ldots,  v_m$ as a host tree. The subtrees corresponding to the clique $C$ are the $2m+1$ paths on the vertices
	\begin{align*}
	\{v_{-m}, \ldots, v_i\} \quad  &\text{ for } \  i = 0, \ldots, m \quad \text{ and } \\
		\{ v_i , \ldots,v_m\} \quad &\text{ for } \ i = -m+1, \ldots, 0.
	\end{align*}
	The common intersection $v_C$ of the clique is the vertex $v_0$.
	Furthermore we define the following subpaths, which are in the periphery of $C$:
	\begin{align*}
	\{v_{-m}, \ldots , v_i\} \quad &\text{ for } \ i = -m, \ldots , -1  \quad \text{and} \\
	\{v_i , \ldots, v_m \} \quad &\text{ for } \ i = 1 , \ldots, m.
	\end{align*}
	For every choice of $i \in \{-m ,\ldots, -1\}$ and $j \in \{1, \ldots, m \}$, we consider the two paths:
	\begin{align*}
		\{v_{-m}, \ldots , v_i\} \quad \text{ and } \quad \{v_j , \ldots ,v_m \}
	\end{align*}
	of the periphery. The anchor set corresponding to this two-element periphery
	set consists of all paths in the host tree
	corresponding to a clique vertex which contain $v_i$ and $v_j$.
	For every choice of $i$ and $j$ these anchor sets differ. Hence there are at least $m^2$ anchor sets.

		Another interesting subclass are the chordal comparability graphs. For these graphs we show a linear bound on the anchor width.
	A graph $G = (V,E)$ is a \emph{comparability graph} if there is a poset $(V,\prec)$ such that two vertices $u,v \in V $ are adjacent in $G$ if and only if $u\prec v$ or $v \prec u $. \\

	\begin{proposition}\label{prop:comparability}
		The anchor width of a chordal comparability graph with $n$ vertices is at most~$2n$.
	\end{proposition}

		\begin{figure}[tb]
		\centering
		\includegraphics[page = 4,scale=0.65]{ComparabilityGraph}
		\caption{Hasse diagram of a poset corresponding to a comparability graph with induced $C_4$ which gives a contradiction in the proof of Proposition \ref{prop:comparability};}
		\label{fig:ComparabilityGraph}
	\end{figure}

	\begin{proof}
		Let $G$ be a chordal comparability graph with corresponding poset $(V, \prec)$.
		Consider a maximal clique $C_{max} = \{c_1, \ldots ,c_m \}$ in $G$.
		A clique in the graph corresponds to a chain in the poset.
		Hence the maximal clique $C_{max}$ of size $m$ corresponds to a maximal chain $c_1 \prec \ldots \prec c_m$ of length $m-1$ in the poset.
		So whenever there is a vertex $v \notin C_{max}$, which is adjacent to $c_i \in C_{max}$ such that $v \prec c_i$ then $v $ is adjacent to $c_k$ for all $k \geq i$. Let $i$ be the minimal element of the clique such that $v \prec c_i$.
		Since the clique is maximal, $i >1$ and $v$ is not comparable to $c_{i-1}$.
		Similar we get for every vertex $w \notin C_{max}$ which is connected to a vertex $c_j$ of the clique with $w  \succ c_j$ that $w$ is connected to all elements $c_k$ of the clique with $k \leq j$.
		Let $c_j$ be the maximal element of the clique connected to $w$, then $j <m$ and $c_{j+1}$ is not comparable to $w$.
		Hence an anchor set in $C_{max}$ is a chain of the form $c_i \prec c_{i+1} \prec \ldots \prec c_{j-1} \prec c_j$.
		Assume there is an anchor set with $1< i <j < m$ and vertices $v$ and $w$ such that $v \prec c_i$ and $w \succ c_j$.
		Then $v$ and $w$ are connected by an edge since $w \succ c_j \succ c_i \succ v$ holds. And since $w$ and $c_{j+1}$ do not share an edge and analogously $v $ and $c_{i-1}$, we get an induced cycle of length $4$ which is not possible since the graph is chordal. In Figure \ref{fig:ComparabilityGraph} the poset is illustrated by its Hasse diagram and gives an illustration of the contradiction.
		This shows that all anchor sets of $C_{max}$ are of the form $c_1 \prec \ldots \prec c_{j-1} \prec c_j$ for $j \leq m$ or $c_i \prec c_{i+1} \prec \ldots \prec  c_m$ for $i \geq 1$. We have at most $2m-1\leq 2n$ possibilities for those sets.
	\end{proof}

	\section{Discussion}
	In this paper we studied an algorithm for computing the neighborhood polynomial of chordal graphs, which is in general an \NP-hard problem. The runtime of the algorithm depends on the introduced parameter anchor width. If the anchor width of a subclass of chordal graphs is bounded, we have a polynomial-time algorithm to compute the neighborhood polynomial.
	In Section~\ref{sec:AnchorWidth} we investigated some subclasses and showed that the anchor width is bounded for chordal graphs with bounded leafage and chordal comparability graphs.
	Furthermore we showed that the anchor width is not bounded for split graphs.
	It would be interesting to get further subclasses of chordal graphs with bounded anchor width.
	It might be possible to give an upper bound for the anchor width using the asteroidal number. In \cite{LinMcKeeWest1998} it is shown that the leafage is an upper bound for the asteroidal number for all chordal graphs and they coincide for chordal graphs whose host tree is a subdivision of $K_{1,n}$ as shown in \cite{Prisner92}.
	Furthermore an infinite family of graphs similar to the one for interval graphs, which shows that the magnitude of the upper bound is optimal, would be interesting.

	On top of that there might be other problems on chordal graphs which are hard in general  but polynomial solvable on those subclasses with bounded anchor width.
	One natural candidate would be graph isomorphism, which is known to be graph-isomorphism-complete on general chordal graphs \cite{LuekerBooth1979} but has recently been shown to be solvable in polynomial time for chordal graphs of bounded leafage \cite{Zeman21}.

	\bibliographystyle{abbrvnat}

redpajama_set_name: RedPajamaArXiv


This Candle Powered Fan uses some small candles to heat one side of a TEC-module, this creates a voltage that will operate the fan. A CPU heatsink is used on the hot and cold side of the module and the result is enough power to spin a small DC fan.
I was wondering where you buy the tec module??


Getting a tattoo is one of the bravest and boldest things that a person can do. It stays on the skin forever and once you get it inked, there is no going back.
There is something mesmerizing with finally tattoo ideas come to life. A tattoo is a form of art and self-expression, and those who have it just seem to stand out from the crowd.
However, no matter how beautiful your tattoo is, sometimes, you just have to cover it up for work or for special occasions.
So, how do you possibly cover a tattoo up? Well, surprise, surprise – you can completely cover it up with makeup! You just have to learn how to do it properly to achieve the best end results. Today, you have come to the right place, as we will teach you how to effectively cover your tattoos with makeup.
First, you have to make sure that your skin is completely clean and free of dirt and oil. Wash it with soap and water and then rinse the area with cold water. This will help close the pores, which is a must as you will be applying tons of makeup in the area.
Once all that's done, make sure to pat it dry with a clean towel or a new piece of tissue.
Choose a cream or liquid based concealer. This will be easier to spread around and can cover the tattoo better.
Make sure to choose a concealer that has very high coverage, as those with only sheer coverage would not do the job. Also, choose a concealer that is at least two to three shades lighter than your skin tone.
Next, grab a makeup sponge and start dabbing the concealer into your skin. Remember to just dab and do not rub, as rubbing will move the product around, leaving your tattoo looking streaky and patchy. Just apply one layer and dab until it looks even.
Let the concealer dry for a few minutes, then follow up with the next step.
After the concealer has completely dried, the next step is the foundation. Again choose a high-coverage foundation such as this Dermacol Make-Up Cover or the Kat Von D Lock-it Foundation. Both have extremely high coverage and can cover your tattoos in just a matter of seconds.
Here is an important tip for you: when buying your foundation and other makeup, make sure to buy only from verified stores and sellers to avoid buying a fake one.
Again, make sure that you buy two to three shades lighter than your skin tone when it comes to the foundation. Just like with the concealer, grab a beauty sponge and apply the foundation with it directly on your skin. Remember to just dab and not rub to avoid moving the foundation around.
Make sure to apply the foundation in thin layers to avoid your makeup from caking. If you need more coverage, wait for the first layer of foundation to dry first, and then follow it up with another thin layer. Build it up one step at a time to make sure you get the coverage that you need.
Now that you have your tattoo covered, it is finally time to set it. Choose a translucent powder instead of one with coverage to avoid making the makeup look cakey.
Next, grab a big, fluffy brush to apply the powder on your skin. Gently dip the brush in the powder and start dabbing it over the concealer and foundation. Again, remember to just dab and not rub the powder to make sure you don't move the concealer and foundation around.
This step will help set the concealer and foundation in place while making the area look like skin again.
Hold on, we are almost done! Just grab a hairspray or a heavy duty makeup setting spray to set your tattoo makeup.
This last step will help make sure that your foundation and concealer won't transfer to anything such as your clothes and other parts of your skin and body.
Hold it at least 10 inches away from your body and start spraying. You only want to spritz it on the area and not drench it completely. Drenching it will make it look patchy and streaky, so make sure to hold it a little far away from your arm tattoo you are trying to cover.
If you are looking for a more permanent solution, then you might want to consider tattoo removal. There are things to consider when getting a tattoo removal, so make sure to keep on reading to know what they are.
Your skin most probably won't go back to its original state. When you get your old tattoo removed and lasered, chances are you are going to get some type of keloid or scar.
Some tattoos will still be partially visible. Remember that laser treatment when it comes to tattoos are subject to how old your tattoo is and how deep it has been inked. The older the tattoo is, the easier it is to remove it.
Tell your doctor if you have any allergies, especially with tattoos, so they would know what type of laser to use.
Different types of lasers can be used for different colors of tattoos.
Before getting a tattoo done, make sure that you really want it and that you are ready for it. Getting a tattoo is a lot of responsibility, so make sure to prepare for it and take care of it properly.


text:
Home/Sports/Sources – Denver Broncos QBs Tested Negative, Will Return Wednesday
Sources – Denver Broncos QBs Tested Negative, Will Return Wednesday
ENGLEWOOD, Colorado – Three quarterbacks of the Denver Broncos, including starter Drew Lock, may return to training and the team's facilities on Wednesday after each test negative Tuesday for COVID-19, multiple sources confirmed to ESPN.
Lock, Brett Rypien and Blake Bortles, who are part of the training squad, were seen as close high-risk contacts with Jeff Driskel, who tested positive for the coronavirus on Thursday. After concerns over contact tracing data and the league's COVID-19 violations and social distancing protocols, the three were removed from the training ground on Saturday and none of them were in uniform for Sunday's 31-3 loss to the New Orleans Saints. It was discovered that the three men had not worn masks around Driskel at one point as they watched a movie together on a day off.
Broncos coach Vic Fangio said on Monday that if the three received negative results on Tuesday they would return to team activities on Wednesday. All three had tested negative again as they had done every day since Driskel was transferred to the reserve / COVID-19 list, sources said.
9News was the first to report that the quarters tested negative again on Tuesday.
Fangio said on Monday that additional discipline is still "on the table" for Lock, Rypien and Bortles if no discipline comes from the NFL, including the possibility of a fine for conduct detrimental to the team.
The Broncos were fined $ 250,000 and Fangio was fined $ 100,000 earlier this season for failing to follow mask protocols and now faces the prospect of a fine as well as a potential loss of a draft for the last incident.
Broncos President and CEO Joe Ellis released a statement Tuesday morning that said, in part, "We share our fans' disappointment for the difficult and unique situation our team faced at the last minute. this week-end. While it's easy to point fingers, all of us need to take responsibility and work together to prevent this from happening again. "
Ellis, who has battled COVID-19 himself this season, also thanked medical staff, players and other team staff. Ellis also said in the statement that he met by video call Monday, president of football operations Broncos / general manager John Elway and Fangio on "what needs to be done to step it up" in the efforts of the COVID-19 team.
Ellis also said that "we will learn from it."
The Broncos have also started the process of signing quarterback Kyle Shurmur, the son of offensive coordinator Pat Shurmur, via COVID-19 protocols. If the Broncos don't release a quarterback by the time Shurmur is cleared to participate in team meetings and practices, he would be the fifth quarterback on the roster or on the practice squad.
Fangio said on Monday that the option to quarantine a quarterback would be discussed for the next few weeks, "but I think if we just follow the protocols we'll be fine."
Lock had apologized on social media ahead of Sunday's game and said he made an "honest mistake, but I'll own."
Practice team receiver Kendall Hinton played 24 of 43 shots of the team in attack in the quarterback on Sunday, while the Broncos had running backs Phillip Lindsay and Royce Freeman the rest of the time. Hinton hit a 1-of-9 pass for 13 yards with two interceptions after his crash course 24 hours on offense, and the Broncos gained 112 total yards.
Islanders 'Ilya Sorokin' hanging out to dry " in their NHL debut
Chilean second assist for Kyle Larson
Bills fans raise money for Lamar Jackson's favorite charity after Ravens QB exit with injury
Report: Former Penn State RB coach Charles Huff to be named Marshall's head coach
Caris LeVert has a mass on the kidney, revealed by MRI after exchange with Pacers; will undergo more tests
With Sean Couturier's injury, Flyers prospect Morgan Frost set to get shot in his natural position

redpajama_set_name: RedPajamaCommonCrawl


text:
Mixed messages
Media messages about bodies & violence
Body-positive Media
GI Joe & Barbie
The Media /
Body-positive Media /
While the mass media all too often present us with oppressive images of ideal beauty and ideal bodies, some media forms are designed to challenge such unrealistic representations. One recent example of this is the Dove Campaign for Real Beauty, that uses the images of women of a variety of ages, ethnicities and body shapes to advertise its products.
Body positivity has been quite a movement that surfaced in the United States around 2012 and promoted the message that "all bodies are beautiful", combating the idea that has bombarded consumers for decades on end that one's body must be stick thin, flawless, and essentially inhuman. With the plethora of media, including social media platforms, many companies, well-known individuals, and everyday people started to post more about the reality of the body ideal and how it is unattainable for anyone.
One company that has taken this movement quite seriously is Aerie. As a clothing, activewear, and intimate brand under American Eagle, Aerie decided to display their products via an extremely diverse group of models. Since 2014, Aerie's media does not include any airbrushing to mask any "imperfections" once deemed unacceptable by the body ideal.
However, the most pressing question is does this actually work? Do women who view ads like Aerie's feel more comfortable with themselves and their bodies? In a 2019 study, the effectiveness of this campaign was assessed by 35 women. The consensus was clear in that the women felt more positively both about Aerie and their bodies compared to other companies' campaigns that haven't adopted the body positive mentality (Rodgers et al., 2019).
Documentaries like Jean Kilbourne's Killing Us Softly are helping to challenge media representations of ideal beauty. However, there's still a long way to go. Kilbourne's career shows this: her first documentary was produced in 1979. Since then, she's made three more updates that show the persistence of this problem -- her most recent, Killing Us Softly 4, was released in 2010. See the full list of films for other titles here.
Last update: 9 April 2021


text:
import * as React from "react";
import { Row, Col, Markdown } from "../../../ui/index";
import { OsUpdateButton } from "./os_update_button";
import { Popover, Position } from "@blueprintjs/core";
import { ColWidth } from "../farmbot_os_settings";
import { FarmbotOsRowProps } from "./interfaces";
import { FbosDetails } from "./fbos_details";
import { t } from "../../../i18next_wrapper";

const getVersionString =
  (fbosVersion: string | undefined, onBeta: boolean | undefined): string => {
    const needsExtension = fbosVersion && !fbosVersion.includes("-") && onBeta;
    const extension = needsExtension ? "-beta" : "";
    return fbosVersion ? fbosVersion + extension : t(" unknown (offline)");
  };

export function FarmbotOsRow(props: FarmbotOsRowProps) {
  const { sourceFbosConfig, dispatch, bot, osReleaseNotes, botOnline } = props;
  const { controller_version, currently_on_beta
  } = bot.hardware.informational_settings;
  const version = getVersionString(controller_version, currently_on_beta);
  return <Row>
    <Col xs={ColWidth.label}>

        {t("FARMBOT OS")}

    </Col>
    <Col xs={3}>
      <Popover position={Position.BOTTOM_LEFT}>

          {t("Version {{ version }}", { version })}

        <FbosDetails
          botInfoSettings={bot.hardware.informational_settings}
          dispatch={dispatch}
          shouldDisplay={props.shouldDisplay}
          sourceFbosConfig={sourceFbosConfig}
          botToMqttLastSeen={props.botToMqttLastSeen}
          timeSettings={props.timeSettings}
          deviceAccount={props.deviceAccount} />
      </Popover>
    </Col>
    <Col xs={3}>
      <Popover position={Position.BOTTOM}>

          {t("Release Notes")}

{props.osReleaseNotesHeading}

          <Markdown>
            {osReleaseNotes}
          </Markdown>

      </Popover>
    </Col>
    <Col xs={3}>
      <OsUpdateButton
        bot={bot}
        sourceFbosConfig={sourceFbosConfig}
        shouldDisplay={props.shouldDisplay}
        botOnline={botOnline} />
    </Col>
  </Row >;
}

redpajama_set_name: RedPajamaGithub


text:
New 2017 Lexus LS to debut at Detroit Motor Show
Flagship fifth-generation luxury Lexus saloon to feature "visionary technology"
by Hugo Griffiths
The new 2017 Lexus LS luxury saloon will be officially unveiled next month at the North American International Auto Show (NAIAS) in Detroit.
Lexus is no stranger to the Detroit motor show: the brand itself launched there in 1989, revealing its first car – the original LS saloon pictured below – to the world. The latest, fifth-generation LS will share underpinnings with the new Lexus LC coupe, which has only recently made its way into the hands of road testers.
The only image of the new LS currently available is a teaser shot, but it's clear from this that the car features Lexus' latest 'spindle' front-end design, seen on the Lexus RC and Lexus NX SUVs – as well as the LC coupe below. Lexus says the LS will feature a "coupe-like silhouette without compromising interior spaciousness".
While scant information is available ahead of the LS' launch, its LC foundations – as well as Lexus' preference for hybrid technology – means we can reasonably expect a 3.5-litre V6 petrol-electric setup to feature. This produces 354bhp in the LC 500h, giving a 0-62mph time of 4.7 seconds and returning around 47mpg. The larger LS is likely to be a heavier car than the LC, though, so expect a 0-62mph time of around six seconds.
It's also possible the 5.0-litre V8 petrol engine of the LC 500 will feature for those who prioritise performance over economy, perhaps in an LS F version.
Being a Lexus means the new LS should feature state-of-the-art technology and luxury. The outgoing LS offers everything from electric window blinds and air-conditioned, reclining, shiatsu-massaging seats to a blu-ray player and 12.3-inch infotainment screen. A tough roster of equipment to top then, but, given Lexus says the new LS will feature "visionary" technology, one that's almost guaranteed to be added to.


Semantic analytics, also termed semantic relatedness, is the use of ontologies to analyze content in web resources. This field of research combines text analytics and Semantic Web technologies like RDF. Semantic analytics measures the relatedness of different ontological concepts.
Some academic research groups that have active project in this area include Kno.e.sis Center at Wright State University among others.
An important milestone in the beginning of semantic analytics occurred in 1996, although the historical progression of these algorithms is largely subjective. In his seminal study publication, Philip Resnik established that computers have the capacity to emulate human judgement. Spanning the publications of multiple journals, improvements to the accuracy of general semantic analytic computations all claimed to revolutionize the field. However, the lack of a standard terminology throughout the late 1990s was the cause of much miscommunication. This prompted Budanitsky & Hirst to standardize the subject in 2006 with a summary that also set a framework for modern spelling and grammar analysis.
In the early days of semantic analytics, obtaining a large enough reliable knowledge bases was difficult. In 2006, Strube & Ponzetto demonstrated that Wikipedia could be used in semantic analytic calculations. The usage of a large knowledge base like Wikipedia allows for an increase in both the accuracy and applicability of semantic analytics.
Given the subjective nature of the field, different methods used in semantic analytics depend on the domain of application. No singular methods is considered correct, however one of the most generally effective and applicable method is explicit semantic analysis (ESA). ESA was developed by Evgeniy Gabrilovich and Shaul Markovitch in the late 2000s. It uses machine learning techniques to create a semantic interpreter, which extracts text fragments from articles into a sorted list. The fragments are sorted by how related they are to the surrounding text.
Latent semantic analysis (LSA) is another common method that does not use ontologies, only considering the text in the input space.
The application of semantic analysis methods generally streamlines organizational processes of any knowledge management system. Academic libraries often use a domain-specific application to create a more efficient organizational system. By classifying scientific publications using semantics and Wikipedia, researchers are helping people find resources faster. Search engines like Semantic Scholar provide organized access to millions of articles.
^ Strube, Michael, and Simone Paolo Ponzetto. "WikiRelate! Computing Semantic Relatedness Using Wikipedia. In Proceedings of the 21st National Conference on Artificial Intelligence, Volume 2, 1419–1424. AAAI'06. Boston, Massachusetts: AAAI Press, 2006.
^ Z. Zhang, A. L. Gentile, and F. Ciravegna, "Recent advances in methods of lexical semantic relatedness – a survey", Natural Language Engineering, vol. 19, no. 04, pp. 411–479, Oct. 2013.
^ Evgeniy Gabrilovich and Shaul Markovitch. 2007. "Computing semantic relatedness using Wikipedia-based explicit semantic analysis". In IJcAI, 1606–1611. Retrieved October 9, 2016.


Luxurious two-bedroom suite with panoramic sea and city views. It includes a modern bathroom with bathtub and shower, elegant living room, dining area and fully functional kitchen with Nespresso coffee machine. Free WiFi is available.


text:
Talk of a coalition with Sinn Féin is nonsensical
Fine Gael TD for Dublin Fingal, Alan Farrell, has described talk of a coalition with Sinn Féin as nonsensical and concerning.
"Putting the history associated with Sinn Féin aside for a moment, their economic policies alone would disqualify them from ever entering government as part of a coalition with Fine Gael. In addition to that, I am of the belief such a proposal would simply not receive support from Fine Gael Parliamentary Party members, our extensive membership across the country or mine, for that matter.
"Sinn Féin's economic policies would be detrimental to our country and, furthermore, given the hypocrisy between the policy they advocate here and those which have been implemented in the North where Sinn Féin have been in a position of government, the simple fact is, we do not know what they would actually do if they ever did make it into a position of authority here.
"Should Sinn Féin ever enter a coalition government in the Republic, I cannot imagine it ever being with Fine Gael. I would say that the likelihood of them making it into government would be at the discretion of others in the opposition benches, specifically, Fianna Fail, should the prospect arise.
"In short, Fine Gael entering coalition with Sinn Féin would be nonsensical both in terms of policy and ideology."


text:
May 31, 2011 Langdon
Beyond Techno-triumphalism
In late May I visited Denton, Texas for the 17th International Conference of the Society for Philosophy and Technology. Speaking on a panel about "The Future of Philosophy and Technology," I offered the following views.
# # # # # # #
Philosophy of Technology in a New Key
By: Langdon Winner
Whatever its particular ideas, claims, arguments, theories, and schools of thought, much of modern philosophy of technology has played its compositions within a particular key signature.
For want of a better term, I would call it the key of Techno-triumphalism. Its familiar themes are progress, advance, conquest, improvement, revolution, prosperity, growth, development, democratization, and two new favorites -- innovation and sustainability.
Over many decades much of the discussion among philosophers has focused upon ways to somehow modify the grand trajectory of triumphalist thought, to correct it when it veered off course.
If only we could move the project at little this way or that way, understand and interpret it more intelligently, then things would be fine. Let's include ordinary people in decision making. Let's emphasize the environment in our schemes for improvement. Let's recognize the voices and contributions of women, of people in developing countries and of others who have traditionally been marginalized.
But the underlying, largely unquestioned, conviction has been that our path surely leads onward and upward. More scientific knowledge along with enhanced philosophical clarity will surely produce better technologies, greater prosperity, and improved prospects for human well-being. Such are the lyrics in classic songs sung in the key of Techno-triumphalism. Unfortunately, these days, the melody sounds discordant, while the words increasingly lack conviction.
The grand chorus among today's universities, tech parks and entrepreneurs is "Celebrate! We're doing innovation!" But if one looks closely, "innovation" is primarily a label for processes and products useful to the world's wealthy few and as money pumps for global corporations. No longer can the great majority of human beings on the planet expect to benefit much, although within the neoliberal refrain, some of the wealth may eventually "trickle down."
Even within the terms of the most optimistic triumphalist projections, current trends are not all that favorable. Some recent analyses of present technical and economic trends take note of an obvious fact – that the 21st century so far has not been a period of world-altering material improvements, but instead an era of technological of stagnation and triviality. (Need I mention flat screen TVs, iPads, Viagra, and other signature products of our era?)
And whatever happened to prosperity from the bubbling cauldrons of nanotech? Where are all those phenomenal high tech industries that were supposed to generate high paying jobs for our children, replacing work lost to global outsourcing?
Along with their fellow citizens, today's philosophers of technology are reluctant to admit that the great festival of the twentieth century (especially in the U.S.A.) – with the automobile, suburb, hyper-consumerism, and the frantic consumption of fossil fuels – is winding down. The era of peak petroleum has probably already arrived and there are simply no good, readily available, cheap substitutes in energy supply. And since our food supply rests squarely on petroleum, this foreshadows continuing crises in supply and cost of food. One can add to this the growing recognition of peak water and peak availability of wide a range of natural resources.
As ominous signs about energy and other material needs accumulate, our leaders in business, government and the academy staunchly refuse to face our predicament squarely, preferring to repeat the beguiling triumphalist tunes of the 19th and 20th centuries.
Perhaps even more significant in the long term is the advent of rapid, possibly irreversible, anthropogenic climate change bringing monster storms, devastating droughts, melting glaciers, rising seas, massive floods, and other disasters.
The good news is that in a bold move to address the problem, the U.S. House of Representatives recently voted on the proposition "climate change is occurring, is caused largely by human activities, and poses significant risks for public health and welfare." The resolution lost 240 to 184.
Reflected in this breathtaking vote is an attitude shared by many of the world's political and business leaders and, alas, quietly shared by many philosophers – a state of complacent denial.
Perhaps the coming "singularity" will save us. Or maybe an unexpected miracle from the research and development labs will confront the combined crises of energy, climate, world hunger, and social unrest, dispatching these maladies with a wave of a magic wand.
My modest suggestion is that a central project for philosophers of technology in the years ahead is to achieve a clear, rational, and hopeful view of human prospects beyond Techno-triumphalism.
There is not enough time today to spell out what this project would entail.
Certainly it would include an awareness of the sweeping consequences of peak energy and climate change. To my way of thinking, it would also involve frank recognition that the core belief of contemporary civilization – that technological development brings economic prosperity, the sure pathway to universal human well-being – is now an exhausted, largely discredited myth.
A realization of this kind need not be occasion for despair. It is clear that humanity must somehow live more lightly on planet Earth, a challenge that necessarily moves questions of social justice to the forefront of concern.
Can the inquiries of philosophers and the anthems of world societies be written in a new key signature?
Strike up the band.
Tags climate, energy, philosophy, technology


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text:
Massacre at Béziers
Part of the Albigensian Crusade
The church of Saint Mary Magdalene where 7,000 people were massacred according to Peter of Vaux-de-Cernay
Béziers, County of Toulouse
22 July 1209 (1209-07-22)
People of Béziers, especially Cathars
Anti-Catharism
Albigensian Crusade
Montségur
The Massacre at Béziers refers to the slaughter of the inhabitants during the sack of Béziers, an event that took place on 22 July 1209, and was the first major military action of the Albigensian Crusade.
2 The sack of Béziers
2.1 "Kill them all, God will know His own"
After Pope Innocent III had declared a crusade to eliminate Catharism in the Languedoc, a crusader army consisting of knights with their retinue (mostly from northern France), professional soldiers, mercenary bands (routiers), and pilgrims, assembled and departed from Lyon in early July 1209.[1] Béziers, a stronghold of Catharism, was the first major town the crusaders encountered on the way to Carcassonne. It was well fortified, amply supplied, and in a position to withstand a long siege. Raymond VI, Count of Toulouse was able to switch sides in time and joined the crusaders at Valence. The attempt by Raymond Roger Trencavel, viscount of Béziers, to peacefully submit was rejected at Montpellier. The viscount departed from Montpellier in a hurry, ahead of the crusader army, to prepare his defenses. On the way to Carcassonne, he stopped at Béziers, promising reinforcements, and taking along some Cathars and Jews.[1]
The sack of Béziers[edit]
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Commanded by the Papal legate, the Abbot of Citeaux, Arnaud Amalric,[2] the crusader army reached the outskirts of Béziers on 21 July 1209. As they started to pitch their camp, the Bishop of Béziers, Renaud de Montpeyroux, tried to avert bloodshed and to negotiate. He came back to Béziers with the message that the town would be spared provided it would hand over their heretics.[3] The bishop had drawn up a list of 222 individuals, mostly Cathars, some Waldensians, likely to be perfecti or leaders of their communities. But in a meeting at the Cathedral, it was determined that to hand over these people was not possible because they had too much support within the town. So the bishop asked the Cathars to leave the town to save themselves. This proposal was rejected, and the bishop left the town with just a few Cathars.[1]
On 22 July the crusaders were busy getting settled and still days away from starting the siege proper. A group of soldiers (perhaps merely armed civilians from the town) made a sortie exiting the gate overlooking the river Orb. As they started to harass routiers and pilgrims of the crusader army, a brawl ensued and soon the attackers from the town found themselves outnumbered and retreated in disarray. The routiers quickly took advantage of the chaos, stormed the walls that were not properly manned, and entered the gate, all without orders. The crusader knights, realizing that the defenses had been broken by the routiers, soon joined the battle, overwhelming the garrison, and the city was doomed.[1]
The routiers rampaged through the streets, killing and plundering, while those citizens who could run sought refuge in the churches – the cathedral the churches of St Mary Magdalene and of St Jude. Yet the churches did not provide safety against the raging mob of invaders. The doors of the churches were broken open, and all inside were slaughtered.
After the massacre it came to the distribution of the city's spoils. The crusader knights became enraged that the rabble of the army had already taken the plunder. They took control of the situation, chased them from the occupied houses and took their booty away. In turn, the angry and disappointed routiers responded by burning down the town. In the engulfing fire, the plunder was lost, and the army left the city in a hurry.
"Kill them all, God will know His own"[edit]
Main article: Caedite eos. Novit enim Dominus qui sunt eius.
Amalric's own version of the siege, described in his letter to Pope Innocent in August 1209 (col.139), states:
Indeed, because there is no strength nor is there cunning against God, while discussions were still going on with the barons about the release of those in the city who were deemed to be Catholics, the servants and other persons of low rank and unarmed attacked the city without waiting for orders from their leaders. To our amazement, crying "to arms, to arms!", within the space of two or three hours they crossed the ditches and the walls and Béziers was taken. Our men spared no one, irrespective of rank, sex or age, and put to the sword almost 20,000 people. After this great slaughter the whole city was despoiled and burnt, as divine vengeance miraculously raged against it.[4]
About twenty years later, Caesarius of Heisterbach relates this story about the massacre,
When they discovered, from the admissions of some of them, that there were Catholics mingled with the heretics they said to the abbot "Sir, what shall we do, for we cannot distinguish between the faithful and the heretics." The abbot, like the others, was afraid that many, in fear of death, would pretend to be Catholics, and after their departure, would return to their heresy, and is said to have replied "Caedite eos. Novit enim Dominus qui sunt eius – Kill them all for the Lord knoweth them that are His" (2 Tim. ii. 19) and so countless number in that town were slain.[5][6]
While there remains doubt that the abbot said these words – also paraphrased as "Kill them all, God will know His own", "Kill them all, God will sort his own", or "Kill them all,/and let God sort them out" – there is little if any doubt that these words captured the spirit of the assault,[7] and that the crusaders intended to kill the inhabitants of a stronghold that offered resistance.[8] However, typically that would involve killing the men, not women and children, and not the clergy. The crusaders allowed the routiers to rampage and kill without restraint, but quickly stepped in when it came to the loot.[1]
It is possible that Amalric's account of the death of 20,000 is exaggerated, just like Peter of Vaux de Cernay's report that 7,000 were slain in the Church of St Magdalene. The town's population at the time is estimated at 10,000–14,500, and an unknown number may have escaped the massacre.[9]
Historian Laurence W. Marvin calls Amalric's exhortation "apocryphal" and "The speed and spontaneity of the attack indicates that the legate may not have actually known what was going on until it was over." Marvin adds "clearly most of Beziers population and buildings survived" and the city "continued to function as a major population center."[10]
Simon de Montfort, who later led the crusade, was a participant in these events but not yet in a leadership role.
The crusaders had achieved a quick and devastating victory. Horror and terror spread through the land, and many castles and towns submitted without further resistance. Carcassonne fell within a month and Raymond-Roger Trencavel died in captivity later that year, his lands being given to de Montfort. However, the crusaders lost the support of the local Catholic population and thus became a hated occupying force.[1] The war became protracted, and eventually the French king entered the war and took control over the Languedoc. The Inquisition then hunted down the remaining Cathars.
During the fire the Cathedral of Saint Nazaire burned and collapsed. A plaque opposite the cathedral records the "Day of Butchery" perpetrated by the "northern barons". A few parts of the Romanesque cathedral survived, and repairs started in 1215. The restoration, along with that of the rest of the city, continued until the 15th century.
^ a b c d e f Zoé Oldenbourg. Massacre at Montségur. A History of the Albigension Crusade (1961). Phoenix, 2006. p. 109ff. ISBN 1-84212-428-5.
^ MD Costen (15 November 1997). The Cathars and the Albigensian Crusade. Manchester University Press 1997. p. 121. ISBN 0-7190-4331-X.
^ Claude Lebédel. Understanding the tragedy of the Cathars. Editions Ouest-France, 2011. p. 59f. ISBN 978-2-7373-5267-6.
^ Albigensian Crusade
^ ExecutedToday.com (22 July 2009). "1209: Massacre of Béziers, "kill them all, let God sort them out"". Retrieved 22 November 2011.
^ "Medieval Sourcebook: Caesarius of Heisterbach: Medieval Heresies: Waldensians, Albigensians, Intellectuals". Fordham.edu. Retrieved 22 November 2011.
^ Russell Jacoby (5 April 2011). Bloodlust: On the Roots of Violence from Cain and Abel to the Present. Free Press, Simon & Schuster. p. 29f. ISBN 978-1-4391-0024-0.
^ William of Tudela, cited in Zoé Oldenburg, Massacre at Montségur, page 116
^ Laurence M. Marvin (25 March 2009). "The Storm of Béziers". Warandgame.com. Archived from the original on April 26, 2012. Retrieved 22 November 2011.
^ Marvin, Laurence W. The Occitan War: A Military and Political History of the Albigensian Crusade, 1209–1218 Cambridge University Press, 2008 p. 43
Archives départementales de l'Hérault, 1209 : le sac de Béziers vu par ses contemporains, dossier préparé par Damien Vaisse, Montpellier, Conseil général de l'Hérault, 2009, 12 p., ill. (édition des témoignages du XIIIe siècle sur le sac de Béziers)
Brenon, Anne. Les Archipels Cathares.
Brenon, Anne. Petit Précis de Catharisme, Loubatières, 1996.
Brenon, Anne. Les cathares : Pauvres du Christ ou apôtres de Satan ?, collection « Découvertes Gallimard » (nº 319), série Religions. Paris: Gallimard, 1997.
Brenon, Anne. Les femmes cathares, Perrin, 1992.
Brenon, Anne, Hérésie, courtoisie et poésie. A la recherche de traces de catharisme dans la littérature occitane du Moyen Âge., in AA.VV. Trobadours et Cathares en Occitanie médiévale; atti del Convegno di Chancelade, 24 e 25 agosto 2002, pp. 61–79.
Dante, Domenico. Il tempo interrotto. Breve storia dei catari in Occidente, Palomar, Bari 2009.
Duvernoy, Jean. Le Catharisme. La religion, 1976.
Duvernoy, Jean. Le Catharisme. L'histoire, 1979.
Duvernoy, Jean. Cathares, Vaudois et Béguins. Dissidents du pays d'Oc, Editions Privat, 1994.
O' Shea, Stephen. The Perfect Heresy: The Life and Death of the Cathars, Profile Books Ltd, 2001.
Coordinates: 43°20′51″N 3°13′08″E / 43.3476°N 3.2190°E / 43.3476; 3.2190
Retrieved from "https://en.wikipedia.org/w/index.php?title=Massacre_at_Béziers&oldid=931258724"
Catharism
Massacres in France
Massacres of Christians
Collective punishment
Persecution of Christian heretics
13th-century Christianity
Articles needing additional references from July 2017


Thank you to everyone who came to Urban Barn this weekend to celebrate our one-year anniversary! It was a blast!
Urban Barn will no longer be open on Sundays starting Easter Sunday, April 23rd. New hours will be Tuesday-Saturday from 10am to 5pm. We will open on Sundays by appointment only. Just let us know when you want to come shopping!
"May happiness attend thy way. This Easter-time and with you to stay."
One Year Down, and Ready to Grow!
Urban Barn is Turning One!


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If you would like your webblog to display the professional plus points, you most likely avoid need the visual aspect of a quite easy handmade web page. Before you produce a web page, you will need someplace to put it. A site can be well improved and can attend when people are searching for information. nevertheless , it can continue being ugly and appearance as a poor supply of details. On the opposing hand a web site can be designed in an attractive design that looks professional. nevertheless , it isn't a website which gets registered in the various search engines like yahoo. The next actions to have a fresh free personal website will be easy and immediate.
You're free to create a site with a considerably more polished, stylish look through the use of software that you may find possibly in stores or on the net. Other reasons to begin a web site include social media and communication. Ahead of you cause a site nevertheless , it is necessary that you just evaluate the current market and confirm demand for the goods or perhaps support. If a web site can be attempting to receive well located for twenty phrases, then simply that web-site will be looking for 20 or more pages. Anyone who's considering your site and wishes to use it just like a source of information will examine your text. Obviously, might likewise build a site of your own to market the own design and style abilities and solutions. Today anybody can make a completely free personal website with their dreams.
If you wish to know anything more then you can certainly see my site, you find almost everything. If you've under no circumstances made a website before, you simply have to find out some basic expertise first. The best method to create a internet site for your property business is always to understand how to do-it-yourself.
People then you do not have a means of locating the site. When you build a website it's important to understand what makes a web site work. A personal website is actually a rather important starting point for every on-line organization.
You've got to supply your website in a particular niche, in order to have an excellent one. This means you have in the end chosen to make a site, but you haven't any concept of where to start! The site should provide a thorough web meeting of who you happen to be and the sort of websites you can easily create. Your initial consideration to determine is what you need your site to contain. The more expensive you are able to purchase your site to rank for sure keywords a lot more free traffic your websites will get. Understanding how to make a web site with WordPress for people who do buiness is really clear-cut.