text:
\section{Introduction}

Tutte's Wheels-and-Whirls Theorem~\cite{tut66} and its extension, Seymour's Splitter Theorem~\cite{sey80}, are inductive tools that have been crucial in the advancement of matroid theory. Since their publication, variants of these theorems have been established whereby additional constraints are imposed on the elements that are available for removal. For example, Oxley et al.~\cite{oxl08} showed that if $B$ is a given basis of a $3$-connected matroid $M$ with no $4$-element fans, and $N$ is a $3$-connected minor of $M$, then, provided a certain necessary condition holds, there is either an element $b\in B$ such that $M/b$ is $3$-connected with an $N$-minor or an element $b^*\in E(M)-B$ such that $M\backslash b^*$ is $3$-connected with an $N$-minor. More recently, Drummond et al.~\cite{dru21} proved a variant of Tutte's Wheels-and-Whirls Theorem for elastic elements. An element $e$ of a $3$-connected matroid $M$ is \emph{elastic} if ${\rm si}(M/e)$, the simplification of $M/e$, and ${\rm co}(M\backslash e)$, the cosimplification of $M\backslash e$, are both $3$-connected. It is shown in~\cite{dru21} that if $|E(M)|\ge 4$, and $M$ has no $4$-element fans and no member of a particular family of $3$-separators, generically referred to as $\Theta$-separators, then $M$ has at least four elastic elements. As a comparison, Bixby's Lemma~\cite{bix82} says that if $e$ is an element of $M$, then either ${\rm si}(M/e)$ or ${\rm co}(M\backslash e)$ is $3$-connected. In this paper, we establish a variant of Seymour's Splitter Theorem for elastic elements. Furthermore, we consider the structure of $M$ if it has exactly four elastic elements, resolve a question of Whittle and Williams~\cite{whi13}, and show that past analogous results, where the removal of elements is relative to a fixed basis, are consequences of~\cite{dru21} and the results in this paper. We next state the main results of the paper.

Let $M$ be a $3$-connected matroid and let $N$ be a $3$-connected minor of $M$. We say that an element $e$ of $M$ is \emph{$N$-elastic} if both ${\rm si}(M/e)$ and ${\rm co}(M\backslash e)$ are $3$-connected and each has an $N$-minor. Furthermore, we say that an element $e$ of $M$ is \emph{$N$-revealing} if
either ${\rm si}(M/e)$ has an $N$-minor and is not $3$-connected, or ${\rm co}(M\backslash e)$ has an $N$-minor and is not $3$-connected.

As well as fans, $\Theta$-separators provide exceptions to the presence of elastic elements. We describe the latter now. For all $n\geq 2$, let $\Theta_n$ denote the matroid whose ground set is the disjoint union $W=\{w_1, w_2, \ldots, w_n\}$ and $Z=\{z_1, z_2, \ldots, z_n\}$, and whose circuits are as follows:
\begin{enumerate}[{\rm (i)}]
\item all $3$-element subsets of $W$;

\item all sets of the form $(Z-\{z_i\})\cup \{w_i\}$, where $i\in \{1, 2, \ldots, n\}$; and

\item all sets of the form $(Z-\{z_i\})\cup\{w_j, w_k\}$, where $i$, $j$, and $k$ are distinct elements of $\{1, 2, \ldots, n\}$.
\end{enumerate}
If $n=2$, then $\Theta_2$ is isomorphic to the direct sum of $U_{1, 2}$ and $U_{1, 2}$, while if $n=3$, then $\Theta_3$ is isomorphic to $M(K_4)$. More generally, it is shown in~\cite{oxl00} that $\Theta_n$ is a matroid for all $n\ge 2$. Furthermore, for the interested reader, $\Theta_n$ is precisely the matroid that underlies the operation of segment-cosegment exchange on $n$ elements~\cite{oxl00}. If $i, j\in \{1, 2, \ldots, n\}$, it is easily checked that $\Theta_n\backslash w_i\cong \Theta_n\backslash w_j$. Up to isomorphism, we denote the matroid $\Theta_n\backslash w_i$ by $\Theta_n^-$. If $n=3$, then $\Theta^-_3$ is isomorphic to a $5$-element fan. We call the elements in $W$ and $Z$ the {\em segment} and {\em cosegment} elements, respectively, of $\Theta_n$ and $\Theta^-_n$.

Now let $M$ be a $3$-connected matroid, and suppose that $r(M)\ge 4$ and $r^*(M)\geq 4$. Let $W$ and $Z$ be rank-$2$ and corank-$2$ subsets of $E(M)$, respectively, such that  $n=\max\{|W|,|Z|\}\geq 3$. We say $W\cup Z$ is a \emph{$\Theta$-separator} of $M$ if either $M|(W\cup Z)$ or $M^*|(W\cup Z)$ is isomorphic to one of the matroids $\Theta_n$ or $\Theta_n^-$. Furthermore, if $N$ is a $3$-connected minor of $M$, we say a $\Theta$-separator \emph{reveals} $N$ in $M$ if either
\begin{enumerate}[(i)]
\item $M|(W\cup Z)\in\{\Theta_n,\Theta_n^-\}$ and at least one element of $Z$ is $N$-revealing in $M$ or, dually,

\item $M^*|(W\cup Z)\in\{\Theta_n,\Theta_n^-\}$ and at least one element of $W$ is $N^*$-revealing in $M^*$.
\end{enumerate}

The following theorem is the above mentioned analogue of Tutte's Wheels-and-Whirls Theorem for elastic elements established in~\cite{dru21}.

\begin{theorem}
\label{elastic2}
Let $M$ be a $3$-connected matroid with no $4$-element fans and no $\Theta$-separators. If $|E(M)|\geq 4$, then $M$ has at least four elastic elements.
\end{theorem}

The main result of this paper is the next theorem.

\begin{theorem}
\label{thm: main}
Let $M$ be a $3$-connected matroid with no $4$-element fans, and let $N$ be a $3$-connected minor of $M$ such that $M$ has no $\Theta$-separators revealing $N$. If $M$ has at least one $N$-revealing element, then $M$ has at least two $N$-elastic elements.
\end{theorem}

\noindent Equivalently, provided the two initial conditions hold, that is $M$ has no $4$-element fans and no $\Theta$-separators revealing $N$, Theorem~\ref{thm: main} says that either $M$ has at least two $N$-elastic elements, or whenever ${\rm si}(M/e)$ has an $N$-minor, then ${\rm si}(M/e)$ is $3$-connected, and whenever ${\rm co}(M\backslash e)$ has an $N$-minor, then ${\rm co}(M\backslash e)$ is $3$-connected. The requirement of Theorem~\ref{thm: main} that $M$ has at least one $N$-revealing element is a necessary one. Consider, for example, when $M$ and $N$ have the same rank. However, the requirement of the two initial conditions in the statement of Theorem~\ref{thm: main} are not completely necessary as shown by the next theorem, a refinement of Theorem~\ref{thm: main}, which describes precisely how fans and $\Theta$-separators may prevent the presence of $N$-elastic elements.

Let $M$ be a matroid on ground set $E$. A $k$-separation $(X, E-X)$ of $M$ is \emph{vertical} if $\min\{r(X), r(E-X)\}\geq k$. Let $(X, \{e\}, Y)$ be a partition of $E(M)$. We say that $(X, \{e\}, Y)$ is a {\em vertical $3$-separation} of $M$ if $(X\cup \{e\}, Y)$ and $(X, Y\cup \{e\})$ are both vertical $3$-separations of $M$, and $e\in {\rm cl}(X)\cap {\rm cl}(Y)$. Furthermore, $Y\cup \{e\}$ is \emph{maximal} if $M$ has no vertical $3$-separation $(X', \{e'\}, Y')$ such that $Y\cup \{e\}$ is a proper subset of $Y'\cup \{e'\}$.
Theorem~\ref{elastic2} is a consequence of~\cite[Theorem 1]{dru21} which says that if $(X, \{e\}, Y)$ is a vertical $3$-separation of a $3$-connected matroid $M$ such that $Y\cup \{e\}$ is maximal, then $X$ contains at least two elastic elements of $M$ unless $X\cup \{e\}$ is a $4$-element fan or $X$ is contained in a $\Theta$-separator. The next theorem extends this latter result to $N$-elastic elements. Its proof is given in Section~\ref{sec: splitter}.

\begin{theorem}
\label{thm: maximal}
Let $M$ be a $3$-connected matroid and let $N$ be a $3$-connected minor of $M$. Let $(X, \{e\}, Y)$ be a vertical $3$-separation of $M$ such that $M/e$ has an $N$-minor and $|X\cap E(N)|\le 1$. If $(X', \{e'\}, Y')$ is a vertical $3$-separation of $M$ such that $Y\cup \{e\}\subseteq Y'\cup \{e'\}$ and $Y'\cup \{e'\}$ is maximal, then $X'$ contains at least two $N$-elastic elements unless $X'\cup \{e'\}$ is a $4$-element fan or $X'$ is contained in a $\Theta$-separator revealing $N$.
\end{theorem}

The elasticity of elements in $\Theta$-separators is described in Section~\ref{sec: prelims}. In particular, a $\Theta$-separator revealing $N$ has at most one $N$-elastic element.
To consider what happens in the statement of Theorem~\ref{thm: maximal} when $X'\cup \{e'\}$ is a $4$-element fan, let $M$ be a $3$-connected matroid. A {\em flower} $\Phi$ is a partition $(P_1, P_2, \ldots, P_n)$ of $E(M)$ such that $|P_i|\ge 2$, and $P_i$ and $P_i\cup P_{i+1}$ are $3$-separating for all $i\in \{1, 2, \ldots, n\}$, where subscripts are interpreted modulo $n$. The parts of $\Phi$ are called {\em petals}. We say $\Phi$ is {\em swirl-like} if $n\ge 4$, and $\sqcap(P_i, P_j)=1$ for all consecutive $i$ and $j$, and $\sqcap(P_i, P_j)=0$ for all non-consecutive $i$ and $j$, where
$$\sqcap(P_i, P_j)=r(P_i)+r(P_j)-r(P_i\cup P_j)$$
is the local connectivity between $P_i$ and $P_j$. For further details regarding flowers, we refer the interested reader to~\cite{oxl04}. The proof of the next theorem is also given in Section~\ref{sec: splitter}.

\begin{theorem}
\label{thm: fans}
Let $M$ be a $3$-connected matroid such that $r(M), r^*(M)\geq 4$, and let $N$ be a $3$-connected minor of $M$. Let $(X, \{e\}, Y)$ be a vertical $3$-separation of $M$ such that $M/e$ has an $N$-minor and $|X\cap E(N)|\le 1$, and let $(X', \{e'\}, Y')$ be a vertical $3$-separation of $M$ such that $Y\cup \{e\}\subseteq Y'\cup \{e'\}$ and $Y'\cup \{e'\}$ is maximal. Suppose that $X'\cup \{e'\}$ is a $4$-element fan $F=(f_1, f_2, f_3, f_4)$, where $e'=f_1$. Then the following hold:
\begin{enumerate}[{\rm (i)}]
\item If $F$ extends to a fan of size at least six, then $F$ contains no elastic elements of $M$.

\item If $F$ extends to a fan of size five but not to a fan of size six, then $F$ contains either exactly one elastic element, namely $f_3$ and this element is $N$-elastic, or $F$ contains no elastic elements.

\item If $F$ does not extend to a fan with more elements, then either it contains exactly two elastic elements, namely $f_2$ and $f_3$, and both are $N$-elastic, or it contains no elastic elements.
\end{enumerate}
Moreover, if $F$ contains no elastic elements, then, up to duality, $M$ has a swirl-like flower $(A, \{f_1, f_2\}, \{f_3,f_4\}, B)$ as shown geometrically in Fig.~\ref{fig: flower}, or there are elements $e$ and $g$ such that $M|(F\cup \{e, g\})\cong M(K_4)$.
\label{fans1}
\end{theorem}

\begin{figure}[ht]
	\centering
	\begin{tikzpicture}[scale=1]
	\coordinate (cntrl1) at (-50:2.5) {};
	\coordinate (cntrl2) at (-20:2.5) {};
	\begin{scope}[every node/.style=element]
	\node (e1) at (-210:2) {};
	\coordinate (c2) at (90:1) {};
	\coordinate (c1) at (30:2);
	\node (e3) at (0,4) {};
	\coordinate (b) at (-120:2) {};

	\coordinate (cntr1) at ($(e1) + (230:1.5)$);
	\coordinate (cntr2) at ($(c1) + (-60:1.5)$);

	\draw (c1) to (e1);
	\node (e2) at ($(e1)!0.5!(e3)$) {};

	\draw (c1) to  (b);
	\draw (e1) to (b);

	\draw (e1) .. controls (cntr1) .. (b);
	\draw (c1) .. controls (cntr2) .. (b);
	\draw (e1) to (e2) to (e3);
	\draw (e2) to (c1);
	\draw (e3) to (c2);

	\node (e4) at (intersection of e3--c2 and e2--c1) {};

	\end{scope}
	\node[scale=1.0] at ($(e1)+(150:0.5)$) {$f_1$};
	\node[scale=1.0] at ($(e2)+(150:0.5)$) {$f_3$};
	\node[scale=1.0] at ($(e3)+(150:0.5)$) {$f_2$};
	\node[scale=1.0] at ($(e4)+(30:0.5)$) {$f_4$};

	\node[scale=1.5] at ($(e1)+(-100:1)$) {$A$};
	\node[scale=1.5] at ($(c1)+(-110:1)$) {$B$};

	\end{tikzpicture}
	\caption{The swirl-like flower $(A, \{f_1, f_2\},\{f_3,f_4\}, B)$ of Lemma \ref{elastic fans}, where, if $|F|=5$, then $f_5$ is an element in $B$.}
	\label{fig: flower}
\end{figure}

Having established a lower bound on the number of $N$-elastic elements by Theorem~\ref{thm: main}, and having a lower bound on the number of elastic elements by Theorem~\ref{elastic2}, it is natural to consider those matroids with the minimum number of such elements.
Let $M$ be a matroid. An exact $3$-separating partition $(X, Y)$ of $E(M)$ is a \emph{sequential $3$-separation} if there is an ordering $(e_1, e_2, \ldots, e_k)$ of $X$ or $Y$ such that $\{e_1, e_2, \ldots, e_i\}$ is $3$-separating for all $i\in\{1, 2, \ldots, k\}$. A matroid has \emph{path-width three} if its ground set is sequential; that is, there is an ordering $(e_1,e_2,\ldots,e_n)$ of its ground set such that $\{e_1,e_2,\ldots,e_i\}$ is $3$-separating for all $i\in \{1,2,\ldots,n\}$. Furthermore, a \emph{path of $3$-separations} in a $M$ is an ordered partition $(P_0, P_1, \ldots, P_k)$ of $E(M)$ with the property that $P_0\cup P_1\cup \cdots \cup P_i$ is exactly $3$-separating for all $i\in \{0, 1, \ldots, k-1\}$. The proofs of the next two theorems are given in Section~\ref{sec: min elts}.

\begin{theorem}
\label{min eltsI}
Let $M$ be a $3$-connected matroid with no $4$-element fans and no $\Theta$-separators. If $M$ has exactly four elastic elements, then $M$ has path-width three.
\end{theorem}

\begin{theorem}
\label{min eltsII}
Let $M$ be a $3$-connected matroid with no $4$-element fans, and let $N$ be a $3$-connected minor of $M$ with $|E(N)|\geq 4$ such that $M$ has no $\Theta$-separators revealing $N$. Let $K$ be the set of $N$-revealing elements of $M$. If $M$ has exactly two $N$-elastic elements $s_1$ and $s_2$, then $K$ has an ordering $(e_1, e_2, \ldots, e_k)$ such that
$$(\{s_1, s_2\}, \{e_1\}, \{e_2\}, \ldots, \{e_k\}, E(M)-K\cup\{s_1, s_2\})$$
is a path of $3$-separations in $M$. Moreover, for every such ordering of $K$, both $M/e_i$ and $M\backslash e_i$ have an $N$-minor for all $i<k$.
\end{theorem}

The results of this paper and~\cite{dru21} have links to the study of matroids in which the removal of elements is relative to a fixed basis~\cite{bre14, oxl08, whi13}. Let $M$ be a $3$-connected matroid, and suppose that we are given a matrix representation of $M$ in standard form relative to some basis $B$. As well as keeping $3$-connectivity, it is often desirable to remove elements from $M$ while keeping the information displayed by this representation. In particular, we want to avoid pivoting. To this end, we can remove elements either by contracting elements in $B$ or deleting elements in $E(M)-B$. Extending the results in~\cite{oxl08}, Whittle and Williams~\cite{whi13} showed that if $|E(M)|\geq 4$ and $M$ has no $4$-element fans, then $M$ has at least four elements $e$ such that either $e\in B$ and ${\rm si}(M/e)$ is $3$-connected, or $e\in E(M)-B$ and ${\rm co}(M\backslash e)$ is $3$-connected. Furthermore, Brettell and Semple~\cite{bre14} gave a splitter theorem analogue of this result. In Section~\ref{sec: fixed basis}, we show that these results are implied by the work of the current paper and its predecessor~\cite{dru21}.

The paper is organised as follows. The next section contains some preliminaries on connectivity and elastic elements. Section~\ref{sec: splitter} consists of the proofs of Theorems~\ref{thm: main}--\ref{fans1}. Section~\ref{sec: min elts} considers those matroids with the minimum number of elastic elements as well as those matroids with the minimum number of $N$-elastic elements, and establishes Theorems~\ref{min eltsI} and~\ref{min eltsII}. Lastly, Section~\ref{sec: fixed basis} contains new proofs of the main results of \cite{bre14} and~\cite{whi13}, and resolves a question posed in~\cite{whi13}. Throughout the paper, the notation and terminology will follow~\cite{ox11}. In addition, if a matroid $M$ has a $\Theta$-separator, then we will implicitly assume that $M$ has rank and corank at least four.

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

\subsection*{Connectivity}

Let $M$ be a matroid with ground set $E$. The \emph{connectivity function} $\lambda_M$ of $M$ is defined on all subsets $X$ of $E$ by
$$\lambda_M(X)=r(X)+r(E-X)-r(M).$$
A subset $X$ of $E$ or a partition $(X, E-X)$ is \emph{$k$-separating} if $\lambda_M(X)\le k-1$ and is {\em exactly $k$-separating} if $\lambda_M(X)=k-1$. A $k$-separating partition $(X, E-X)$ is a \emph{$k$-separation} if $\min \{|X|, |E-X|\}\geq k$. A matroid is \emph{$n$-connected} if it has no $k$-separations for all $k < n$. Furthermore, two $k$-separations $(X_1, Y_1)$ and $(X_2, Y_2)$ of $M$ are said to \emph{cross} if each of the intersections $X_1\cap Y_1$, $X_1\cap Y_2$, $X_2\cap Y_1$, and $X_2\cap Y_2$ is non-empty. The next lemma is a particularly useful tool for handling crossing separations and follows from the fact that the connectivity function of a matroid is submodular. We refer to an application of this lemma as \emph{by uncrossing}.

\begin{lemma}
\label{uncrossing}
Let $M$ be a $k$-connected matroid, and let $X$ and $Y$ be $k$-separating subsets of $E(M)$.
\begin{enumerate}[{\rm (i)}]
\item If $|X\cap Y|\geq k-1$, then $X\cup Y$ is $k$-separating.

\item If $|E(M)-(X\cup Y)|\geq k-1$, then $X\cap Y$ is $k$-separating.
\end{enumerate}
\end{lemma}

The following three lemmas are used throughout the paper. The first follows from orthogonality. The second is a consequence of the first, while the third is a consequence of the first and second.

\begin{lemma}
\label{orthogonality}
Let $(X, \{e\}, Y)$ be a partition of the ground set of a matroid $M$. Then $e\in {\rm cl}(X)$ if and only if $e\not\in {\rm cl}^*(Y)$.
\end{lemma}

\begin{lemma}
\label{3sep1}
Let $X$ be an exactly $3$-separating set in a $3$-connected matroid $M$, and suppose that $e\in E(M)-X$. Then $X\cup \{e\}$ is $3$-separating if and only if $e\in {\rm cl}(X)\cup {\rm cl}^*(X)$.
\end{lemma}

\begin{lemma}
Let $(X, Y)$ be an exactly $3$-separating partition of a $3$-connected matroid $M$, and suppose that $|X|\ge 3$ and $e\in X$. Then $(X-\{e\}, Y\cup \{e\})$ is exactly $3$-separating if and only if $e$ is in exactly one of ${\rm cl}(X-\{e\})\cap {\rm cl}(Y)$ and ${\rm cl}^*(X-\{e\})\cap {\rm cl}^*(Y)$.
\label{3sep2}
\end{lemma}

\subsection*{Vertical and cyclic connectivity}

Recalling the terminology and notation from the introduction, a $k$-separation $(X, Y)$ of a matroid $M$ is \emph{vertical} if $\min\{r(X), r(Y)\}\geq k$. A partition $(X, \{e\}, Y)$ of $E(M)$ is a \emph{vertical $3$-separation} of $M$ if $(X\cup \{e\}, Y)$ and $(X, Y\cup \{e\})$ are both vertical $3$-separations of $M$ and $e\in {\rm cl}(X)\cap {\rm cl}(Y)$. Furthermore, $Y\cup \{e\}$ is {\em maximal} in this separation if there is no vertical $3$-separation $(X', \{e'\}, Y')$ of $M$ such that $Y\cup \{e\}$ is a proper subset of $Y'\cup \{e'\}$. A $k$-separation $(X, Y)$ of $M$ is {\em cyclic} if both $X$ and $Y$ contain circuits. Vertical $k$-separations and cyclic $k$-separations are dual concepts. In particular, it is straightforward to show (for example, see \cite[Lemma 9]{dru21}) that if $(X, Y)$ is a partition of the ground set of a $k$-connected matroid $M$, then $(X,Y)$ is a cyclic $k$-separation of $M$ if and only if $(X, Y)$ is a vertical $k$-separation of $M^*$. As such, we say a partition $(X, \{e\}, Y)$ of the ground set of a $3$-connected matroid $M$ is a \emph{cyclic $3$-separation} if $(X, \{e\}, Y)$ is a vertical $3$-separation of $M^*$.

The first of the next two lemmas indicates why vertical $3$-separations and cyclic $3$-separations arise in the context of non-elastic elements. In combination with the duality link between vertical $3$-separations and cyclic $3$-separations, the first lemma is a straightforward strengthening of~\cite[Lemma~3.1]{oxl08} and the second lemma follows from Lemma~\ref{3sep2}.

\begin{lemma}
\label{vertical1}
Let $M$ be a $3$-connected matroid, and suppose that $e\in E(M)$. Then ${\rm si}(M/e)$ is not $3$-connected if and only if $M$ has a vertical $3$-separation $(X, \{e\}, Y)$. Dually, ${\rm co}(M\backslash e)$ is not $3$-connected if and only if $M$ has a cyclic $3$-separation $(X, \{e\}, Y)$.
\end{lemma}

\begin{lemma}
\label{vertical2}
Let $M$ be a $3$-connected matroid. If $(X, \{e\}, Y)$ is a vertical $3$-separation of $M$, then $(X -{\rm cl}(Y), \{e\}, {\rm cl}(Y)-\{e\})$ is also a vertical $3$-separation of $M$. Dually, if $(X, \{e\}, Y)$ is a cyclic $3$-separation of $M$, then $(X-{\rm cl}^*(Y), \{e\}, {\rm cl}^*(Y)-\{e\})$ is also a cyclic $3$-separation of $M$.
\end{lemma}

\subsection*{Fans, segments, and \texorpdfstring{$\Theta$}{Theta}-separators}

Let $M$ be a $3$-connected matroid. A subset $F$ of $E(M)$ with at least three elements is a \emph{fan} if there is an ordering $(f_1, f_2, \ldots, f_k)$ of the elements of $F$ such that
\begin{enumerate}[(i)]
\item for all $i\in \{1, 2, \ldots, k-2\}$, the triple $\{f_i, f_{i+1}, f_{i+2}\}$ is either a triangle or a triad, and

\item for all $i\in \{1, 2, \ldots, k-3\}$, if $\{f_i, f_{i+1}, f_{i+2}\}$ is a triangle, then $\{f_{i+1}, f_{i+2}, f_{i+3}\}$ is a triad, while if $\{f_i, f_{i+1}, f_{i+2}\}$ is a triad, then $\{f_{i+1}, f_{i+2}, f_{i+3}\}$ is a triangle.
\end{enumerate}
If $k\geq 4$, then the elements $f_1$ and $f_k$ are the \emph{ends} of $F$. Furthermore, if $\{f_1, f_2, f_3\}$ is a triangle, then $f_1$ is a {\em spoke-end}; otherwise, $f_1$ is a {\em rim-end}. It is elementary to show that if $F=(f_1, f_2, f_3, f_4)$ is a $4$-element fan with spoke-end $f_1$ in a $3$-connected matroid $M$ of rank at least four, then $(\{f_2, f_2, f_4\}, \{f_1\}, E(M)-F)$ is a vertical $3$-separation of $M$ in which $E(M)-\{f_2, f_3, f_4\}$ is maximal. The next result is \cite[Lemma~13]{dru21} and details when elements of a $4$-element fan are elastic.
\begin{lemma}
\label{elastic fans}
Let $M$ be a $3$-connected matroid such that $r(M),r^*(M)\geq 4$, and let $F=(f_1, f_2, \ldots, f_n)$ be a maximal fan of $M$.
\begin{enumerate}[{\rm (i)}]
\item If $n\ge 6$, then $F$ contains no elastic elements of $M$.

\item If $n=5$, then $F$ contains either exactly one elastic element, namely $f_3$, or no elastic elements of $M$.

\item If $n=4$, then $F$ contains either exactly two elastic elements, namely $f_2$ and $f_3$, or no elastic elements of $M$.
\end{enumerate}
Moreover, if $n\in \{4, 5\}$ and $F$ contains no elastic elements, then, up to duality, $M$ has a swirl-like flower $(A, \{f_1, f_2\}, \{f_3,f_4\}, B)$ as shown geometrically in Fig.~\ref{fig: flower}, or $n=5$ and there is an element $g$ such that $M|(F\cup\{g\})\cong M(K_4)$.
\end{lemma}

A \emph{segment} of a matroid $M$ is a subset $X$ of $E(M)$ such that $M|X$ is isomorphic to a rank-$2$ uniform matroid. The next lemma is elementary and is used repeatedly in this paper.

\begin{lemma}
\label{segdelete}
Let $M$ be a $3$-connected matroid and let $L$ be a segment of $M$ with at least four elements. If $\ell\in L$, then $M\backslash \ell$ is $3$-connected.
\end{lemma}

Recall the definition of a $\Theta$-separator given in the introduction. The next four lemmas are extracted from~\cite{dru21}. The first lemma follows from the proofs of~\cite[Lemma~13]{dru21} and~\cite[Lemma~16]{dru21}, while the second lemma is an immediate consequence of the first. The third lemma is~\cite[Theorem~1]{dru21}. The fourth lemma follows by combining~\cite[Lemma~18]{dru21} and~\cite[Lemma~19]{dru21}.

\begin{lemma}
Let $M$ be a $3$-connected matroid, and let $S$ be a $\Theta$-separator of $M$ such that $M|S$ is isomorphic to either $\Theta_n$ or $\Theta^-_n$, where $n\ge 3$. Let $W$ and $Z$ be the set of segment and cosegment elements of $M|S$, respectively. If $w\in W$, then ${\rm si}(M/w)$ is not $3$-connected. Furthermore, if  $z\in Z$, then ${\rm co}(M\backslash z)$ is not $3$-connected, unless there is no element $x\in {\rm cl}(W)$ such that $(Z-\{z\})\cup \{x\}$ is a circuit.
\label{exception}
\end{lemma}

\begin{lemma}
\label{theta elastic}
Let $M$ be a $3$-connected matroid and let $S$ be a $\Theta$-separator of $M$. If $M|S\cong \Theta_n$ for some $n\geq 3$, then no element of $S$ is elastic. If $M|S\cong \Theta_n^-$ for some $n\geq 3$, then $S$ contains a unique elastic element unless there exists an element $e\in E(M)-S$ such that $M|(S\cup\{e\})\cong \Theta_n$.
\end{lemma}

\begin{lemma}
\label{elastic1}
Let $M$ be a $3$-connected matroid with a vertical $3$-separation $(X, \{e\}, Y)$ such that $Y\cup \{e\}$ is maximal. If $X\cup \{e\}$ is not a $4$-element fan and $X$ is not contained in a $\Theta$-separator, then at least two elements of $X$ are elastic.
\end{lemma}

\begin{lemma}
Let $M$ be a $3$-connected matroid with a vertical $3$-separation $(X, \{e\}, Y)$ such that $Y\cup \{e\}$ is maximal. If $X$ is contained in a $\Theta$-separator $S$, then either
\begin{enumerate}[{\rm (i)}]
\item $X$ is a rank-$3$ cocircuit, $M^*|S$ is isomorphic to either $\Theta_n$ or $\Theta^-_n$, where $n=|X\cup \{e\}|-1$, and there is a unique element $x\in X$ such that $x$ is a segment element of $M^*|S$ and $(X-\{x\})\cup \{e\}$ is the set of cosegment elements of $M^*|S$, or

\item $X\cup \{e\}$ is a circuit, $M|S$ is isomorphic to either $\Theta_n$ or $\Theta^-_n$ for some $n\in \{|X|, |X|+1\}$, and $X$ is a subset of the cosegment elements of $M|S$.
\end{enumerate}
\label{lem: thetamax}
\end{lemma}

\subsection*{Separations and minors}

The following two lemmas concern $3$-connected minors across $2$-separations. The first is elementary, and the second is a slight strengthening of \cite[Lemma 4.5]{bre14} and follows from the proof of that lemma.

\begin{lemma}
\label{2sep}
Let $(X, Y)$ be a $2$-separation of a connected matroid $M$, and let $N$ be a $3$-connected minor of $M$. Then $\{X, Y\}$ has a member $U$ such that $|U\cap E(N)|\leq 1$. Moreover, if $u\in U$, then
\begin{enumerate}[{\rm (i)}]
\item $M/u$ has an $N$-minor if $M/u$ is connected, and
\item $M\backslash u$ has an $N$-minor if $M\backslash u$ is connected.
\end{enumerate}
\end{lemma}

\begin{lemma}
Let $M$ be a $3$-connected matroid, and let $N$ be a $3$-connected minor of $M$. Let $(X,\{e\},Y)$ be a vertical $3$-separation of $M$ such that $M/e$ has an $N$-minor, where $|X\cap E(N)|\leq 1$ and $Y\cup \{e\}$ is closed. Then $M/x$ has an $N$-minor for every element $x$ of $X$, and there is at most
one element of $X$, say $x'$, such that $M\backslash x'$ has no $N$-minor. Moreover, if such an element $x'$ exists, then $x'\in {\rm cl}^*(Y)$ and $e\in {\rm cl}(X-\{x'\})$.
\label{BS lem 4.5+}
\end{lemma}

We end the preliminaries by considering equivalent conditions for when a $\Theta$-separator reveals a $3$-connected minor.

\begin{lemma}
\label{theta reveal}
Let $M$ be a $3$-connected matroid such that $r(M),r^*(M)\geq 4$, and let $N$ be a $3$-connected minor of $M$. Let $S\subseteq E(M)$ such that $M|S$ is isomorphic to either $\Theta_n$ or $\Theta^-_n$, where $n\ge 3$. Suppose that $W$ and $Z$ are the sets of segment and cosegment elements of $M|S$, respectively. Then the following statements are equivalent:
\begin{enumerate}[{\rm (i)}]
\item At least one element of $Z$ is $N$-revealing in $M$.

\item The cosimplification ${\rm co}(M\backslash z)$ has an $N$-minor for at least two elements $z\in Z$.

\item  Both ${\rm si}(M/z)$ and ${\rm co}(M\backslash z)$ have an $N$-minor for all $z\in Z$ and ${\rm co}(M\backslash w)$ has an $N$-minor for all $w\in W$.
\end{enumerate}
Moreover, if $|E(N)|\leq 3$, then {\rm(i)--(iii)} always hold.
\end{lemma}

\begin{proof}
Since $M|S$ is isomorphic to either $\Theta_n$ or $\Theta_n^-$, there is a labelling of the elements $w_1, w_2, \ldots, w_k$ of $W$ and $z_1, z_2, \ldots, z_n$ of $Z$ such that $(Z-\{z_i\})\cup \{w_i\}$ is a circuit of $M$ for all $i\in \{1, 2, \ldots, k\}$, where $k\in \{n, n-1\}$. Now (iii) certainly implies (ii). Assume that (ii) holds, and let $i\in \{1, 2, \ldots, k\}$. It is straightforward to observe that, as $r(M), r^*(M)\geq 4$, the partition
$$((Z-\{z_i\})\cup \{w_i\}, \{z_i\}, E(M)-(Z\cup \{w_i\}))$$
of $E(M)$ is a cyclic $3$-separation of $M$. Thus, by Lemma~\ref{vertical1}, ${\rm co}(M\backslash z_i)$ is not $3$-connected, and so (ii) implies (i).

We next show that (i) implies (iii) when $|E(N)|\geq 4$. Let $i\in\{1, 2, \ldots, k\}$, and suppose that ${\rm co}(M\backslash z_i)$ has an $N$-minor. Since $N$ is simple and $Z-\{z_i\}$ is a series class of $M\backslash z_i$, and so $(Z-\{z_i\})\cup \{w_i\}$ is $2$-separating in $M\backslash z_i$, it follows that $|(Z\cup\{w_i\})\cap E(N)|\leq 1$. Therefore, by the dual of Lemma~\ref{BS lem 4.5+}, $M\backslash w_i$ has an $N$-minor, and both $M\backslash z_j$ and $M/z_j$ have an $N$-minor for all $j\in \{1, 2, \ldots, n\}-\{i\}$. In particular, $z_j$ is $N$-revealing for all $j\in \{1, 2, \ldots, k\}$. Thus, the initial choice of $i\in \{1, 2, \ldots, k\}$ was arbitrary. Hence both $M/z$ and $M\backslash z$ have an $N$-minor for all $z\in Z$, and $M\backslash w$ has an $N$-minor for all $w\in W$. Thus (iii) holds, and so (i)--(iii) are equivalent if $|E(N)|\ge 4$.

We complete the proof by showing, without assuming (i), that if $|E(N)|\le 3$, then (iii) holds and so, by above, (ii) and (i) also hold. To do this, it suffices to show that, for all $z \in Z$ and $w\in W$, each of ${\rm co}(M\backslash z)$, ${\rm si}(M/z)$, and ${\rm co}(M\backslash w)$ has a $U_{1, 3}$- and $U_{2, 3}$-minor. By Lemma~\ref{exception} and Bixby's Lemma, ${\rm si}(M/z)$ and ${\rm co}(M\backslash w)$ are both $3$-connected. Furthermore, ${\rm si}(M/z)$ has rank at least three and ${\rm co}(M\backslash w)$ has corank at least three. It now follows that ${\rm si}(M/z)$ and ${\rm co}(M\backslash w)$ each have $U_{1, 3}$ and $U_{2, 3}$ as minors. To show that ${\rm co}(M\backslash z)$ has a $U_{1, 3}$- and a $U_{2, 3}$-minor, it suffices to show that, as ${\rm co}(M\backslash z)$ is connected, it has rank and corank at least two. Since $r^*(M)\geq 4$, ${\rm co}(M\backslash z)$ has corank at least three. As $z$ is in at least one circuit of the form $(Z-\{z_i\})\cup \{w_i\}$ in $M$, we have by orthogonality with these circuits that any triad containing $z$ must either be contained in $Z$ or contain an element of $W$. It follows that $z$ is in at most one triad of $M$ with an element outside of $Z$. In particular, ${\rm co}(M\backslash z)$ has rank at least $r(M)-(n-2)=r(M\backslash Z)$ when $n\geq 4$ and rank at least $r(M)-2$ when $n=3$. Thus ${\rm co}(M\backslash z)$ has rank at least two, completing the proof of the lemma.
\end{proof}

A consequence of the last lemma is that every $\Theta$-separator reveals each of the matroids $U_{0, 0}$, $U_{0, 1}$, $U_{1, 1}$, $U_{1, 2}$, $U_{1, 3}$, and $U_{2, 3}$. We freely use this fact throughout the remainder of the paper.

\section{Proofs of Theorems~\ref{thm: main}, \ref{thm: maximal}, and~\ref{fans1}}
\label{sec: splitter}

In this section we prove Theorems~\ref{thm: main}--\ref{fans1} . We begin with three lemmas. The first two lemmas concern elastic elements in matroids with rank and corank at least four.

\begin{lemma}
\label{smallN1}
Let $M$ be a $3$-connected matroid such that $r(M), r^*(M)\ge 4$, and let $N$ be a $3$-connected minor of $M$ with at most three elements. Then every elastic element of $M$ is $N$-elastic.
\end{lemma}

\begin{proof}
Let $x$ be an elastic element of $M$. Then ${\rm si}(M/x)$ and ${\rm co}(M\backslash x)$ are both $3$-connected. Furthermore, as $r(M), r^*(M)\ge 4$, we have that ${\rm si}(M/x)$ has rank at least three and ${\rm co}(M\backslash x)$ has corank at least three. Thus, as ${\rm si}(M/x)$ is $3$-connected, ${\rm si}(M/x)$ contains a circuit, but ${\rm si}(M/x)$ is not a circuit, and so ${\rm si}(M/x)$ has a $U_{2, 3}$- and a $U_{1, 3}$-minor. By duality, ${\rm co}(M\backslash x)$ also has a $U_{1, 3}$- and a $U_{2, 3}$-minor. As $|E(N)|\le 3$, it follows that $N$ is a minor of either $U_{1, 3}$ or $U_{2, 3}$, and the lemma follows.
\end{proof}

Theorem~\ref{fans1} is an immediate consequence of Lemma~\ref{elastic fans} and the next lemma.

\begin{lemma}
\label{lem: N-elastic}
Let $M$ be a $3$-connected matroid of corank at least four, and let $N$ be a $3$-connected minor of $M$. Let $(X,\{e\},Y)$ be a vertical $3$-separation of $M$ such that $M/e$ has an $N$-minor and $|X\cap E(N)|\leq 1$. If $Y\cup \{e\}$ is closed, then every elastic element in $X$ is $N$-elastic.
\end{lemma}

\begin{proof}
Let $x$ be an elastic element of $X$. If $|E(N)|\leq 3$, then, by Lemma~\ref{smallN1}, $x$ is $N$-elastic. Thus we may assume that $|E(N)|\ge 4$. In particular, $N$ is simple and cosimple, and so if $M/x$ or $M\backslash x$ has an $N$-minor, then ${\rm si}(M/x)$ and ${\rm co}(M\backslash x)$ has an $N$-minor, respectively. Therefore, by Lemma~\ref{BS lem 4.5+}, $x$ is $N$-elastic unless $x$ is the unique exception in the statement of Lemma~\ref{BS lem 4.5+}, in which case, $x\in {\rm cl}^*(Y)$ and $e\in {\rm cl}(X-\{x\})$. Suppose that $x$ is this unique exception. Then, as $x\in {\rm cl}^*(Y)$, it follows by Lemma~\ref{orthogonality}, that $x\not\in {\rm cl}(X-\{x\})$. Therefore, as $(Y\cup \{e\}, X)$ is a $3$-separation of $M$, we have
\begin{align*}
2 & = r(Y\cup \{e\})+r(X)-r(M) \\
& = r(Y\cup \{e\})+r(X-\{x\})+1-r(M).
\end{align*}
In particular,
$$1 = r(Y\cup \{e\})+r(X-\{x\})-r(M\backslash x),$$
and so $(Y\cup \{e\}, X-\{x\})$ is a $2$-separation of $M\backslash x$. Since $e\in {\rm cl}(X-\{x\})$, the partition $(Y, (X-\{x\})\cup \{e\})$ is also a $2$-separation of $M\backslash x$. Now, as $x$ is elastic, ${\rm co}(M\backslash x)$ is $3$-connected, and so at least one of $Y\cup \{e, x\}$ and $X$ has corank~$2$, and at least one of $Y\cup \{x\}$ and $X\cup \{e\}$ has corank~$2$. By Lemma~\ref{orthogonality}, $e\not\in {\rm cl}^*(X)\cup {\rm cl}^*(Y)$. Thus $r^*(X)=r^*(Y\cup \{x\})=2$. But then, as $M$ is $3$-connected, $r^*(M)=3$, contradicting the assumption that $M$ has corank at least four. Hence $x$ is not the exception, and the lemma holds.
\end{proof}

\begin{figure}[ht]
	\begin{tikzpicture}[scale=1.5]
	\begin{scope}[every node/.style=element]

	\node (x3) at (60:2) {};
	\node (x2) at (60:1) {};
	\node (x1) at (0,0) {};
	\node (y1) at (1,0) {};
	\node (y2) at (2,0) {};
	\node (y3) at (3,0) {};
	\node (x4) at (60:3) {};
	\node (e) at (intersection of  x2--y2 and y1--x3) {};
	\draw (x4) to (x3) to (x2) to (x1) to (y1) to (y2) to (y3);
	\draw (y2) to (e) to (x2);
	\draw (y1) to (e) to (x3);
	\end{scope}
	\begin{scope}
	\node[scale=1.0] at ($(x1)+(180:0.30)$) {$x_1$};
	\node[scale=1.0] at ($(x2)+(180:0.30)$) {$x_2$};
	\node[scale=1.0] at ($(x3)+(180:0.30)$) {$x_3$};
	\node[scale=1.0] at ($(x4)+(180:0.30)$) {$x_4$};
	\node[scale=1.0] at ($(e)+(30:0.25)$) {$e$};
	\node[scale=1.0] at ($(y1)+(-90:0.3)$) {$y_1$};
	\node[scale=1.0] at ($(y2)+(-90:0.3)$) {$y_2$};
	\node[scale=1.0] at ($(y3)+(-90:0.3)$) {$y_3$};
	\end{scope}
	\end{tikzpicture}
	\caption{The $3$-connected matroid $L_8$.}
	\label{fig: L8}
\end{figure}

The condition in the statement of Lemma~\ref{lem: N-elastic} that $M$ has corank at least four is necessary. To see this, let $L_8$ denote the $3$-connected rank-$3$ matroid shown in Fig.~\ref{fig: L8}. Let $X=\{x_1, x_2, x_3, x_4\}$ and $Y=\{y_1, y_2, y_3\}$. Then $(X, \{e\}, Y)$ is a cyclic $3$-separation of $L_8$, and $L_8\backslash e$ has a $U_{2, 4}$-minor whose ground set contains $Y$. The element $x_1$ of $L_8$ is elastic but it is not $U_{2, 4}$-elastic. However, every element of $X-\{x_1\}$ is $U_{2,4}$-elastic. The next lemma captures this last observation and is the corank-three analogue of Lemma~\ref{lem: N-elastic}.

\begin{lemma}
\label{lem: corank3}
Let $M$ be a $3$-connected rank-$3$ matroid, and let $N$ be a $3$-connected minor of $M$. Let $(X,\{e\},Y)$ be a cyclic $3$-separation of $M$ such that $M\backslash e$ has an $N$-minor and $|X\cap E(N)|\leq 1$. If $X\cup \{e\}$ is not a $4$-element fan, then there is at most one element of $X$ that is not $N$-elastic. Moreover, if such an element $x$ exists, then $x\in {\rm cl}(Y)$.
\end{lemma}

\begin{proof}
Since $M$ is $3$-connected and $(X, \{e\}, Y)$ is a cyclic $3$-separation of $M$, we have $r(X)=r(Y)=2$, and $|X|, |Y|\ge 3$. Furthermore, $|{\rm cl}(Y)\cap X|\le 1$. Now, as $M\backslash e$ has an $N$-minor and $|X\cap E(N)|\leq 1$, it follows that $N$ is a minor of $U_{2, n}$, where $n=|{\rm cl}(Y)|\le |Y|+1$. Let $x\in X-{\rm cl}(Y)$. As $X\cup\{e\}$ is not a $4$-element fan, $|X-{\rm cl}(Y)|\ge 3$, and so ${\rm co}(M\backslash x)=M\backslash x$ and $M\backslash x$ is $3$-connected. Furthermore, ${\rm si}(M/x)$ is isomorphic to either $U_{2, n}$ or $U_{2, n+1}$ depending on whether or not $e$ is in a triangle with $x$. In particular, ${\rm si}(M/x)$ and ${\rm co}(M\backslash x)$ are both $3$-connected with $N$-minors. This completes the proof of the lemma.
\end{proof}

We next prove Theorem~\ref{thm: maximal}.

\begin{proof}[Proof of Theorem~\ref{thm: maximal}]
Let $(X, \{e\}, Y)$ be a vertical $3$-separation of $M$ such that $M/e$ has an $N$-minor and $|X\cap E(N)|\le 1$. Without loss of generality, we may assume that $Y\cup\{e\}$ is closed. Now let $(X', \{e'\}, Y')$ be a vertical $3$-separation of $M$ such that $Y\cup \{e\}\subseteq Y'\cup \{e'\}$ and $Y'\cup \{e'\}$ is maximal, and suppose that $X'\cup \{e'\}$ is not a $4$-element fan. If $r^*(M)=3$, then, by Lemma~\ref{lem: corank3}, $X'$ contains at least two $N$-elastic elements. Thus, we may assume that $r^*(M)\ge 4$. Assume that $X'$ is contained in a $\Theta$-separator $S$. If $|E(N)|\leq 3$, then $S$ reveals $N$. Say $|E(N)|\geq 4$, in which case, $N$ is simple and cosimple. By Lemma~\ref{lem: thetamax}, either
\begin{enumerate}[(I)]
\item $M^*|S$ is isomorphic to either $\Theta_n$ or $\Theta^-_n$, where $n=|X'\cup \{e'\}|-1$, and there is a unique element $x\in X'$ such that $x$ is a segment element of $M^*|S$ and $(X'-\{x\})\cup \{e'\}$ is the set of cosegment elements of $M^*|S$, or

\item $X'\cup \{e'\}$ is a circuit, $M|S$ is isomorphic to either $\Theta_n$ or $\Theta^-_n$ for some $n\in \{|X'|, |X'|+1\}$, and $X'$ is a subset of the cosegment elements of $M|S$.
\end{enumerate}
If (I) holds, then, for all $x'\in X'-\{x\}$, it follows by Lemma~\ref{BS lem 4.5+} that $M/x'$, and hence ${\rm si}(M/x')$, has an $N$-minor. Thus Lemma~\ref{theta reveal}(ii) holds, and so $S$ reveals $N$. If (II) holds, then, by Lemma~\ref{BS lem 4.5+}, there are at least two elements $x'\in X'$ such that $M\backslash x'$, and hence ${\rm co}(M\backslash x')$, has an $N$-minor. Again, we deduce by Lemma~\ref{theta reveal} that $S$ reveals $N$. Thus we may assume that $X'$ is not contained in any $\Theta$-separator. Then, as $Y'\cup\{e'\}$ is maximal, it follows by Lemma~\ref{elastic1} that $X'$ contains at least two elastic elements. By Lemma~\ref{lem: N-elastic}, each of these elastic elements is $N$-elastic, thereby completing the proof of the theorem.\end{proof}

Lastly, we use Theorem~\ref{thm: maximal} to prove Theorem~\ref{thm: main}.

\begin{proof}[Proof of Theorem~\ref{thm: main}]
Let $e$ be an $N$-revealing element of $M$. Then, up to duality, ${\rm si}(M/e)$ has an $N$-minor and is not $3$-connected. It follows by Lemmas~\ref{vertical1} and \ref{2sep} that $M$ has a vertical $3$-separation $(X, \{e\}, Y)$ such that $|X\cap E(N)|\leq 1$. Choosing $(X-{\rm cl}(Y), \{e\}, {\rm cl}(Y)-\{e\})$ if necessary, $M$ has a vertical $3$-separation $(X', \{e'\}, Y')$ such that $Y\cup \{e\}\subseteq Y'\cup \{e'\}$ and $Y'\cup \{e'\}$ is maximal. Since $M$ has no $4$-element fans or $\Theta$-separators revealing $N$, it follows by Theorem~\ref{thm: maximal} that $X'$ contains at least two $N$-elastic elements, completing the proof of the theorem.
\end{proof}

\section{Matroids with the Smallest Number of Elastic Elements}
\label{sec: min elts}

In this section, we prove Theorems~\ref{min eltsI} and~\ref{min eltsII}. We begin with three lemmas. The first lemma follows easily from the definitions and the second is \cite[Lemma 6.3]{bre14}. A proof of the third lemma can be found in~\cite{whi99}.

\begin{lemma}
\label{BS lem 6.1}
Let $(X, Y)$ be a partition of the ground set of a matroid $M$ such that $|X|, |Y|\geq 2$. Then $(X, Y)$ is a sequential $3$-separation of $M$ if and only if for some $U\in \{X, Y\}$, there is a path of $3$-separations $(P_0, P_1, \ldots, P_k, U)$ in $M$ such that $|P_0|=2$ and $|P_i|=1$ for all $i\in\{1, 2, \ldots, k\}$.
\end{lemma}

\begin{lemma}
\label{BS lem 6.3}
Let $M$ be a $3$-connected matroid with distinct elements $s_1$ and $s_2$. Let $U$ be a subset of $E(M)-\{s_1, s_2\}$ such that $|E(M)-(U\cup\{s_1, s_2\})|\geq 2$. If, for each $u\in U$, there is a path of $3$-separations $(X_u, \{u\}, Y_u)$ in $M$ such that $\{s_1, s_2\}\subseteq X_u\subseteq U\cup \{s_1, s_2\}$, then there is an ordering $(u_1, u_2, \ldots, u_k)$ of $U$ such that
$$(\{s_1, s_2\}, \{u_1\}, \{u_2\}, \ldots, \{u_k\}, E(M)-(U\cup\{s_1, s_2\}))$$
is a path of $3$-separations in $M$.
\end{lemma}

\begin{lemma}
\label{triangle}
Let $C^*$ be a rank-$3$ cocircuit of a $3$-connected matroid $M$. If $e\in C^*$ has the property that ${\rm cl}_M(C^*)-\{e\}$ contains a triangle of $M/e$, then ${\rm si}(M/e)$ is $3$-connected
\end{lemma}

We now prove Theorem~\ref{min eltsI}.
\begin{proof}[Proof of Theorem~\ref{min eltsI}]
Suppose that $M$ has exactly four elastic elements $f_1$, $f_2$, $g_1$, and $g_2$. Then, for all $e\in E(M)-\{f_1, f_2, g_1, g_2\}$ either ${\rm si}(M/e)$ is not $3$-connected or ${\rm co}(M\backslash e)$ is not $3$-connected. Thus, by Lemmas~\ref{BS lem 6.1}, \ref{BS lem 6.3} and \ref{vertical1} it suffices to show that there is a partition, say $(\{f_1,f_2\},\{g_1,g_2\})$, of the set of elastic elements of $M$ such that every vertical $3$-separation or cyclic $3$-separation of $M$ is of the form $(X,\{e\},Y)$, where $\{f_1, f_2\}\subsetneqq X$ and $\{g_1, g_2\}\subsetneqq Y$. Suppose that this fails. Then there exists two partitions $(X_1, \{e_1\}, Y_2)$ and $(X_2, \{e_2\}, Y_2)$ of $E(M)$ such that each partition is either a vertical or a cyclic $3$-separation of $M$, and each of the intersections $X_1\cap X_2$, $X_1\cap Y_2$, $Y_1\cap X_2$, and $Y_1\cap Y_2$ contains a unique elastic element. Without loss of generality, we may assume that $e_1\in Y_2$, $e_2\in X_1$, $f_1\in X_1\cap X_2$, $f_2\in X_1\cap Y_2$, $g_1\in Y_1\cap X_2$ and $g_2\in Y_1\cap Y_2$, and that, up to duality, $(X_1, \{e_1\}, Y_1)$ is a vertical $3$-separation.

By uncrossing $X_2\cup\{e_2\}$ with each of $X_1$ and $X_1\cup\{e_1\}$, we have that $(Y_1\cap Y_2)\cup\{e_1\}$ and $Y_1\cap Y_2$ are $3$-separating. If $r(Y_1\cap Y_2)\geq 3$, then $(Y_1\cap Y_2, \{e_1\}, X_1\cup X_2)$ is a vertical $3$-separation of $M$ and so, by Lemma~\ref{elastic1}, $Y_1\cap Y_2$ contains at least two elastic elements, a contradiction. We deduce that $r((Y_1\cap Y_2)\cup\{e_1\})=2$. Furthermore, if $Y_1\cap X_2=\{g_1\}$, then, as $M$ has no $4$-element fans, $|Y_1\cap Y_2|\ge 3$ and $Y_1\cap Y_2$ contains a cycle. Also, as $e_1\in {\rm cl}(X_1)$, we have that $X_1\cup \{e_1\}$ contains a cycle, and so $(Y_1-\{g_1\}, \{g_1\}, X_1\cup\{e_1\})$ is a cyclic $3$-separation of $M$, contradicting the fact that $g_1$ is elastic. Thus $|Y_1\cap X_2|\geq 2$.

Next, by uncrossing $Y_1$ with each of $X_2$ and $X_2\cup\{e_2\}$, we see that $(X_1\cap Y_2)\cup\{e_1, e_2\}$ and $(X_1\cap Y_2)\cup\{e_2\}$ are exactly $3$-separating. If $r((X_1\cap Y_2)\cup \{e_2\})\geq 3$, then $((X_1\cap Y_2)\cup \{e_2\}, \{e_1\}, X_2\cup Y_1)$ is a vertical $3$-separation of $M$ and it follows by Lemma~\ref{elastic1} that $X_1\cap Y_2$ contains at least two elastic elements, a contradiction. We deduce that $r((X_1\cap Y_2)\cup \{e_1, e_2\})=2$. In particular, $e_2\in {\rm cl}(\{e_1, f_2\})$, and so $(X_2, \{e_2\}, Y_2)$ is a vertical $3$-separation of $M$. By Lemma~\ref{vertical2}, we may assume that $X_2\cup \{e_2\}$ is closed.

If $Y_1\cap Y_2=\{g_2\}$, then either $Y_2\cup\{e_2\}$ is a $4$-element fan, a contradiction, or $(Y_2-\{g_2\}, \{g_2\}, X_2\cup \{e_2\})$ is a cyclic $3$-separation of $M$, contradicting the fact that $g_2$ is elastic. Hence $|Y_1\cap Y_2|\geq 2$. Lastly, if $X_1\cap Y_2=\{f_2\}$, then $(Y_2-\{f_2\}, \{f_2\}, X_2\cup \{e_2\})$ is a cyclic $3$-separation of $M$, another contradiction. Thus, $|X_1\cap Y_2|\geq 2$ and the set $(X_1\cap Y_2)\cup\{e_1,e_2\}$ is a segment of at least four elements. As $X_2\cup \{e_2\}$ is closed, it follows that $Y_2$ is a rank-$3$ cocircuit and so, as $|Y_1\cap Y_2|\ge 3$, Lemmas~\ref{segdelete} and~\ref{triangle} imply that every element of $X_1\cap Y_2$ is elastic. This final contradiction completes the proof.
\end{proof}

The next lemma is used in the proof of Theorem~\ref{robust}. It is the analogue of Theorem~\ref{min eltsII} for when the size of $E(N)$ is at most three.

\begin{proposition}
\label{smallN2}
Let $M$ be a $3$-connected matroid with no $4$-element fans and no $\Theta$-separators, and let $N\in\{U_{0, 1}, U_{1, 1}, U_{1, 2}, U_{1, 3}, U_{2, 3}\}$. If $r(M), r^*(M)\geq 3$ and $|E(M)|\geq 8$, then $M$ has at least four $N$-elastic elements. Moreover, if $M$ has exactly four $N$-elastic elements, then $M$ has path-width three.
\end{proposition}

\begin{proof}
By Theorem~\ref{elastic2}, $M$ has at least four elastic elements. If every elastic element of $M$ is $N$-elastic, then the proposition follows from Theorem~\ref{min eltsI}. Thus we may assume that $M$ has an elastic element $e$ which is not $N$-elastic. By Lemma~\ref{smallN1}, we may assume that, up to duality, $r(M)=3$. Since $|E(M)|\geq 8$, it follows that $r^*(M)\geq 5$, and so the $3$-connected matroid ${\rm co}(M\backslash e)$ has corank at least four and rank at least two. In particular, ${\rm co}(M\backslash e)$ has a $U_{1, 3}$- and a $U_{2, 3}$-minor, and therefore an $N$-minor. Hence, as $e$ is not $N$-elastic, we have that ${\rm si}(M/e)$ has no $N$-minor. This is only possible if ${\rm si}(M/e)\cong U_{2,3}$ and $N\cong U_{1,3}$, in which case $M$ is comprised of a triangle, say $\{e_1, e_2, e_3\}$, and three segments $L_1$, $L_2$, and $L_3$ such that $L_1\cap L_2\cap L_3=\{e\}$ and $e_i\in L_i$ for all $i\in \{1, 2, 3\}$. As $M$ has at least eight elements, at least one of these segments, say $L_1$, has at least four elements. A routine check shows that every element of $E(M)-L_1$ as well as at least one element of $L_1$ is $U_{2, 4}$-elastic, and thus $U_{1, 3}$-elastic and $U_{2, 3}$-elastic. Hence $M$ has at least five $N$-elastic elements, completing the proof of the lemma.
\end{proof}

To see that the requirement of Proposition~\ref{smallN2} that $M$ have rank and corank at least three is necessary, consider the case when $M$ is isomorphic to $U_{2,  5}$ and $N$ is isomorphic to $U_{1, 3}$. If $e\in E(M)$, then $M/e$, which is isomorphic to $U_{1, 4}$, has a $U_{1, 3}$-minor but ${\rm si}(M/e)$, which is isomorphic to $U_{1, 1}$, has no $U_{1, 3}$-minor. Thus, in this case, $M$ has no $N$-elastic elements. To see that the requirement $|E(M)|\geq 8$ is necessary, consider the case when $M$ is isomorphic to $F_7$ and $N$ is isomorphic to $U_{1, 3}$. If $e\in E(M)$, then $M/e$ has a $U_{1, 3}$-minor but ${\rm si}(M/e)$, which is isomorphic to $U_{2, 3}$, has no $U_{1, 3}$-minor. Thus, again, $M$ has no $N$-elastic elements.

We next prove Theorem~\ref{min eltsII}.

\begin{proof}[Proof of Theorem~\ref{min eltsII}]
Suppose that $M$ has exactly two $N$-elastic elements $s_1$ and $s_2$. We first show that $|E(M)-(K\cup \{s_1, s_2\})|\geq 2$. If $M$ has no $\Theta$-separators, then, by Theorem~\ref{elastic2}, $M$ has at least four elastic elements and so $|E(M)-(K\cup \{s_1, s_2\})|\ge 2$. Therefore assume that $M$ has a $\Theta$-separator. Let $W$ be a rank-$2$ subset and $Z$ a corank-$2$ subset of $E(M)$ such that $W\cup Z$ is a $\Theta$-separator of $M$. Since $M$ has no $4$-element fans, it follows that $\min\{|W|, |Z|\}\geq 3$. By Lemma~\ref{theta elastic}, at most one element of $W\cup Z$ is elastic. Therefore, if $|E(M)-(K\cup \{s_1, s_2\})|\leq 1$, then at least one element of $W$ and at least one element of $Z$ is $N$-revealing. As $M$ has no $\Theta$-separators revealing $N$, we deduce that $|E(M)-(K\cup \{s_1, s_2\})|\geq 2$.

Next, for each $e\in K$, we select a certain path of $3$-separations $(X_e, \{e\}, Y_e)$. Let $e\in K$. Up to duality, we may assume that ${\rm si}(M/e)$ has an $N$-minor and is not $3$-connected. Then, by Lemmas~\ref{vertical1}, \ref{vertical2} and~\ref{2sep}, there is a vertical $3$-separation $(X, \{e\}, Y)$ of $M$ such that $Y\cup \{e\}$ is closed and $|X\cap E(N)|\leq 1$. By Theorem~\ref{thm: maximal}, $X$ contains at least two $N$-elastic elements. Thus $\{s_1, s_2\}\subseteq X$. Furthermore, by Lemma~\ref{BS lem 4.5+}, $M/x$ has an $N$-minor for all $x\in X$ and there is at most one element, $x'$ say, for which $M\backslash x'$ has no $N$-minor. If there is no such element $x'$, then let $X_e=X$ and $Y_e=Y$. Otherwise, we note by Lemma~\ref{BS lem 4.5+}, that $x'\in {\rm cl}^*(Y)$ and thus $(X-\{x'\},\{e\},Y\cup\{x'\})$ is a path of $3$-separations by Lemma~\ref{3sep1}. In this case, let $X_e=X-\{x'\}$ and $Y_e=Y\cup \{x'\}$. Observe that, by this selection process, $\{s_1,s_2\}\in X_e$ and, for all $x\in X_e$, both $M/x$ and $M\backslash x$ have an $N$-minor. Moreover, as $|E(N)|\geq 4$, the latter property implies that $X_e \subseteq K\cup\{s_1,s_2\}$.

Now let $Y=E(M)-(K\cup\{s_1, s_2\})$. By an application of Lemma~\ref{BS lem 6.3}, there is an ordering $(e_1, e_2, \ldots, e_k)$ of $K$ such that $(\{s_1, s_2\}, \{e_1\}, \{e_2\}, \ldots, \{e_k\}, Y)$ is a path of $3$-separations in $M$. It remains to show that $M\backslash e_i$ and $M/e_i$ have an $N$-minor for all $i < k$. Dualising if necessary, we may assume that $e_k\in {\rm cl}(Y)$. We may also assume that $k\geq 2$.

\begin{sublemma}
Let $A=\{s_1, s_2, e_1, e_2, \ldots, e_k\}$ and let $L={\rm cl}(Y)-Y$. If $|L|\ge 2$, then $M\backslash \ell$ and $M/\ell$ have an $N$-minor for all $\ell\in L$.
\label{sub}
\end{sublemma}

Since $e_k\in L$, it follows by submodularity that
\begin{align*}
1\le r(L)=r({\rm cl}(Y)\cap A)\leq r(Y) + r(A) - r(M)=2.
\end{align*}
Let $\ell\in L$, and suppose that $|L|\ge 2$. Now either $(A-\{\ell\}, \{\ell\}, Y)$ is a vertical $3$-separation, or at least one of $A$ and $Y$ has rank two. Thus, by a combination of Lemma~\ref{vertical1} and Bixby's Lemma or by Lemma~\ref{segdelete}, the matroid ${\rm co}(M\backslash \ell)$ is $3$-connected. Therefore, by the definition of $K$, the matroid ${\rm si}(M/\ell)$, and hence $M/\ell$, has an $N$-minor. If either $A$ or $Y$ has rank two, then, as $N$ is simple and $M/\ell$ has an $N$-minor, $M\backslash \ell'$ has an $N$-minor for all $\ell'\in L-\{\ell\}$. Furthermore, if $(A-\{\ell\}, \{\ell\}, Y)$ is a vertical $3$-separation, then $(A-\{\ell\}, Y)$ is a $2$-separation of $M/\ell$ in which $\ell'\in {\rm cl}(A-\{\ell, \ell'\})\cap {\rm cl}(Y)$ for all $\ell'\in L-\{\ell\}$. Since $N$ is $3$-connected, either $|(A-\{\ell\})\cap E(N)|\le 1$ or $|Y\cap E(N)|\le 1$. Also, as $M/\ell/\ell'$ is not connected, $M/\ell\backslash \ell'$ is connected. Thus, by Lemma~\ref{2sep}, $M\backslash \ell'$ has an $N$-minor for all $\ell'\in L-\{\ell\}$. Since the choice of $\ell$ was arbitrary, we deduce that if $|L|\ge 2$, then $M\backslash \ell$ and $M/\ell$ have an $N$-minor for all $\ell\in L$.

If $Y$ spans $E(M)$, then, as $k\ge 2$, it follows by (\ref{sub}) that $M\backslash e_i$ and $M/e_i$ have an $N$-minor for all $i < k$. Thus we may suppose that $Y$ does not span $E(M)$. Let $j$ be the highest index such that $e_j\not\in {\rm cl}(Y)$. Let $A_j=\{s_1, s_2, e_1, e_2, \ldots, e_j\}$ and let $B_j=\{e_{j+1}, e_{j+2}, \ldots, e_k\}\cup Y$. If $A_j$ is independent, then, as $r(A_j)+r(B_j)-r(M)=2$, it follows that $r^*(A_j)=2$. In this case, as $M\backslash s_1$ has an $N$-minor, $M/e_i$ has an $N$-minor for all $i\le j$. Furthermore, if $|A_j|\ge 2$, then, by the dual of Lemma~\ref{segdelete}, $M/e_i$ is $3$-connected for all $i\le j$. So, by the definition of $K$, the matroid $M\backslash e_i$ has an $N$-minor for all $i\le j$. If $|A_j|=1$, then $j=1$, $\{s_1, s_2, e_1\}$ is a triad of $M$, and $e_k\in {\rm cl}(\{s_1, s_2, e_1\})$. Since $M$ has no $4$-element fans, it follows by Lemma~\ref{triangle} that $M/e_1$ is $3$-connected. Thus, again by the definition of $K$, the matroid $M\backslash e_1$ has an $N$-minor. Therefore, we may assume that $A_j$ is dependent. If $A_j-\{e_j\}$ is independent, then $e_j\in {\rm cl}(A_j-\{a_j\})$ and $e_j\in {\rm cl}^*(B_j)$, contradicting orthogonality. Therefore $A_j-\{a_j\}$ is dependent and so, as $B_j$ is dependent, $\{A_j-\{e_j\}, \{e_j\}, B_j\}$ is a cyclic $3$-separation of $M$. Since ${\rm co}(M\backslash e_j)$ is not $3$-connected, it follows by Bixby's Lemma and the definition of $K$ that $M\backslash e_j$ has an $N$-minor. If $|B_j\cap E(N)|\le 1$, then, by the dual of Theorem~\ref{thm: maximal}, $B_j$ contains two $N$-elastic elements. But $M$ has exactly two $N$-elastic elements $s_1$ and $s_2$, and $\{s_1, s_2\}\subseteq A_j$. Hence $|(A_j-\{e_j\})\cap E(N)|\le 1$.

Now let $L'={\rm cl}^*(B_j)-B_j$. If $|L'|\ge 2$, then, by a similar, but dual, argument used to established~(\ref{sub}), $M\backslash \ell'$ and $M/\ell'$ have an $N$-minor for all $\ell'\in L'$. Furthermore, if $|L'|\ge 2$, then $|{\rm cl}(A_j-L')\cap {\rm cl}(B_j)|=0$, while if $|L'|=1$, then $|{\rm cl}(A_j-\{e_j\})\cap {\rm cl}(B_j)|\le 1$. If, for some $i\in \{1, 2, \ldots, k-1\}$, we have $\{e_i\}={\rm cl}(A_j-\{e_j\})\cap {\rm cl}(B_j)$, then, as $e_i\in {\rm cl}(Y)$, it follows by~(\ref{sub}) that $M\backslash e_i$ and $M/e_i$ have an $N$-minor. Thus, by the dual of Lemma~\ref{BS lem 4.5+}, to complete the proof of the theorem it suffices to show that $M/e_j$ has an $N$-minor when $|L'|=1$. Since ${\rm si}(M/e_j)$ is $3$-connected, $M/e_j/e_k$ is connected. Therefore, as $(A_j, Bj-\{e_k\})$ is a $2$-separation of $M/e_k$, Lemma~\ref{2sep} implies that $M/e_j$ has an $N$-minor.
\end{proof}

\section{Applications to Fixed-Basis Theorems}
\label{sec: fixed basis}

In this section, we show that two previous results concerning the removal of elements relative to a fixed basis are consequences of the results in this paper and~\cite{dru21}. Moreover, we also resolve a question posed in~\cite{whi13}. Let $M$ be a $3$-connected matroid and let $B$ be a basis of $M$. Following \cite{whi13}, we say that an element $e$ of $M$ is \emph{removable with respect to $B$} if either
\begin{enumerate}[{\rm (i)}]
\item $e\in B$ and ${\rm si}(M/e)$ is $3$-connected, or

\item $e\in E(M)-B$ and ${\rm co}(M\backslash e)$ is $3$-connected.
\end{enumerate}
Of course, if an element is elastic, then it is removable with respect to any basis. However, using an appropriate choice for a basis $B$, it is easily checked that a $5$-element fan may have no removable elements with respect to $B$. Nevertheless, removable elements are abundant in all larger $\Theta$-separators. The straightforward proof of the following lemma is omitted.

\begin{lemma}
\label{theta A}
Let $M$ be a $3$-connected matroid and let $B$ be a basis of $M$. Let $W$ be a rank-$2$ subset and let $Z$ be a corank-$2$ subset of $M$ such that $W\cup Z$ is a $\Theta$-separator of $M$ with at least six elements. Then
\begin{enumerate}[{\rm (i)}]
\item $|{\rm cl}(W)-B|\geq |{\rm cl}(W)|-2$ and ${\rm co}(M\backslash w)$ is $3$-connected for all $w\in {\rm cl}(W)$, and

\item $|B \cap {\rm cl}^*(Z)|\geq |{\rm cl}^*(Z)|-2$ and ${\rm si}(M/z)$ is $3$-connected for all $z\in {\rm cl}^*(Z)$.
\end{enumerate}
\end{lemma}

We now show that the main result of \cite{whi13} follows from Theorems~\ref{elastic2} and~\ref{min eltsI}, and a treatment of $\Theta$-separators.

\begin{theorem}[\cite{whi13}, Theorem 1.1]
Let $M$ be a $3$-connected matroid with no $4$-element fans, where $|E(M)|\geq 4$. Let $B$ be a basis of $M$. Then $M$ has at least four elements that are removable with respect to $B$. Moreover, if $M$ has exactly four removable elements with respect to $B$, then $M$ has path-width three.
\end{theorem}

\begin{proof}
First 	suppose that there is a rank-$2$ subset $W$ and a corank-$2$ subset $Z$ of $E(M)$ such that $W\cup Z$ is a $\Theta$-separator of $M$, in which case, $r(M), r^*(M)\geq 4$. Up to duality, we may assume that $M|(W\cup Z)\in \{\Theta_n,\Theta_n^-\}$. As $M$ has no $4$-element fans, $n\ge 4$. By Lemma~\ref{theta A}, at least $|Z|-2$ elements of $Z$ and at least $|W|-2$ elements of $W$ are removable with respect to $B$. If $Z$ spans $M$, then $|{\rm cl}(W)|\geq 4$ and, as $|B\cap {\rm cl}(W)|\leq 2$, it follows by Lemma~\ref{segdelete} that $M$ has at least four elements that are removable with respect to $B$. Moreover, in this instance, $M$ has path-width three as a sequential ordering of $E(M)$ is obtained by first progressing through the elements in $Z$, and then through the elements in ${\rm cl}(W)$. Thus we may assume that $Z$ does not span $M$. Then, for any $w\in W$, the partition
$$((W\cup Z)-\{w\}, \{w\}, E(M)-(W\cup Z))$$
is a vertical $3$-separation of $M$. Let $(U,\{e\},V)$ be a vertical $3$-separation of $M$ such that $V\cup\{e\}$ is maximal and contains $W\cup Z$. Then, by Lemma~\ref{elastic1}, $U$ has at least two elastic elements, or it is contained in a $\Theta$-separator. In the latter case, by combining Lemmas~\ref{lem: thetamax} and~\ref{theta A}, we see that the set $E(M)-(W\cup Z)$ has at least two elements that are removable with respect to $B$ and so, in both cases, $M$ has at least five elements that are removable with respect to $B$.

To complete the proof, we may now assume that $M$ has no $\Theta$-separators. Then, by Theorem~\ref{elastic2}, $M$ has at least four elastic elements, and so $M$ has at least four elements that are removable with respect to $B$. Moreover, if $M$ has exactly four removable elements with respect to $B$, then these are precisely the elastic elements of $M$, and thus $M$ has path-width~$3$ by Theorem~\ref{min eltsI}.
\end{proof}

In~\cite{whi13}, Whittle and Williams asked if there exists a $3$-connected matroid $M$ with no $4$-element fans such that for every basis $B$ of $M$ there are exactly four elements of $M$ which are removable with respect to $B$. In particular, the next proposition shows that no matroid with at least four elements has this property.

\begin{proposition}
Let $M$ be a $3$-connected matroid with no $4$-element fans such that $|E(M)|\geq 4$. Then there exists a basis $B$ of $M$ such that $M$ has at least five removable elements with respect to $B$.
\end{proposition}

\begin{proof}
First suppose that $M$ has a rank-$2$ subset $W$ and a corank-$2$ subset $Z$ such that $W\cup Z$ is a $\Theta$-separator of $M$. By duality, we may assume that $M|(W\cup Z)\in\{\Theta_n,\Theta_n^-\}$. Since $M$ has no $4$-element fans, $n\geq 4$. Let $B$ be a basis of $M$ containing the independent set $Z$. Then, by Lemma~\ref{theta A}, every element of $Z$ and at least one element of $W$ is removable with respect to $B$, giving a total of at least five such elements. Thus, we may assume $M$ has no $\Theta$-separators. Then, by Theorem~\ref{elastic2}, $M$ has at least four elastic elements. Since elastic elements are removable with respect to any basis, the proposition holds if $M$ has at least five elastic elements. So assume that $M$ has exactly four elastic elements. The only $3$-connected matroid on four elements is $U_{2, 4}$, which has a $4$-element fan. Thus $M$ has at least five elements. Let $e$ be a non-elastic element of $M$. As $M$ has no coloops, $M$ has a basis containing $e$ as well as a basis avoiding $e$. If $e$ is not removable with respect to any basis, it follows that the matroids ${\rm si}(M/e)$ and ${\rm co}(M\backslash e)$ are not $3$-connected, a contradiction to Bixby's Lemma. Hence $e$ is removable with respect to some basis $B$, in which case, $M$ has at least five removable elements with respect to $B$.
\end{proof}

Let $M$ be a $3$-connected matroid, let $N$ be a $3$-connected minor of $M$, and let $B$ be a basis of $M$. Following \cite{bre14}, an element $e$ of $M$ is called \emph{$(N,B)$-robust} if either
\begin{enumerate}[{\rm (i)}]
\item $e\in B$ and $M/e$ has an $N$-minor, or

\item $e\in E(M)-B$ and $M\backslash e$ has an $N$-minor.
\end{enumerate}
Furthermore, such an element is called \emph{$(N,B)$-strong} if either
\begin{enumerate}[{\rm (i)}]
\item $e\in B$ and ${\rm si}(M/e)$ is $3$-connected with an $N$-minor, or

\item $e\in E(M)-B$ and ${\rm co}(M\backslash e)$ is $3$-connected with an $N$-minor.
\end{enumerate}

Evidently, an $N$-elastic element of $M$ is $(N,B)$-strong for every basis $B$ of $M$. The next lemma follows by combining Lemma~\ref{theta A} with Lemma~\ref{theta reveal}.

\begin{lemma}
\label{theta B}
Let $M$ be a $3$-connected matroid, let $N$ be a $3$-connected minor of $M$, and let $B$ be a basis of $M$. Let $S$ be a $\Theta$-separator of $M$ with at least six elements. If $S$ reveals $N$ in $M$, then at least $|S|-4$ elements of $S$ are $(N,B)$-strong.
\end{lemma}

We end the paper by showing that the two main results of~\cite{bre14} follow from Theorems~\ref{thm: main} and~\ref{min eltsII}, and a treatment of $\Theta$-separators.

\begin{theorem}[\cite{bre14}, Theorems 1.1 and 1.2]
Let $M$ be a $3$-connected matroid with no $4$-element fans such that $|E(M)|\geq 5$. Let $N$ be a $3$-connected minor of $M$, and let $B$ be a basis of $M$.
\begin{enumerate}[{\rm (i)}]
\item If $M$ has two distinct $(N, B)$-robust elements, then $M$ has two distinct $(N,B)$-strong elements.

\item Let $P$ denote the set of $(N, B)$-robust elements of $M$. If $M$ has precisely two $(N, B)$-strong elements, then $(P, E(M)-P)$ is a sequential $3$-separation of $M$.
\end{enumerate}
\label{robust}
\end{theorem}

\begin{proof}
If $M$ has rank or corank at most two, then the theorem follows easily from the fact that $|E(M)|\geq 5$. Furthermore, a routine check shows that the theorem holds if $|E(M)|\in \{6, 7\}$. Thus we may assume that $r(M), r^*(M)\geq 3$ and $|E(M)|\geq 8$. Since $M$ has no $4$-element fans, any $\Theta$-separator of $M$ has at least seven elements. Therefore, if $M$ has a $\Theta$-separator revealing $N$, then, by Lemma~\ref{theta B}, $M$ has at least three $(N, B)$-strong elements. Thus we may assume that $M$ has no such $\Theta$-separators. We first prove (i). If $|E(N)|\leq 3$, then, by Proposition~\ref{smallN2}, $M$ has at least four $N$-elastic elements, and so (i) holds. Thus we may assume that $|E(N)|\geq 4$, in which case, every $(N, B)$-robust element is either $(N, B)$-strong or $N$-revealing. It follows that either each of the two guaranteed $(N,B)$-robust elements are $(N,B)$-strong or, by Theorem~\ref{thm: main}, $M$ has at least two $N$-elastic elements. In either instance, $M$ has at least two $(N,B)$-strong elements, thereby proving (i).

To prove (ii), suppose that $M$ has precisely two $(N, B)$-strong elements $\{s_1, s_2\}$, in which case, $|E(N)|\ge 4$. If $M$ has no $N$-revealing elements, then $P=\{s_1, s_2\}$ and $(P, E(M)-P)$ is trivially a sequential $3$-separation of $M$. So assume that $M$ has at least one $N$-revealing element. In this case, it follows by Theorem~\ref{thm: main} that $s_1$ and $s_2$ are $N$-elastic, and that $M$ has no further $N$-elastic elements. Now let $K$ be the set of $N$-revealing elements of $M$. Note that $P-\{s_1,s_2\}\subseteq K$. By Theorem~\ref{min eltsII}, $K$ has an ordering $(e_1, e_2, \ldots, e_k)$ such that
$$(\{s_1, s_2\}, \{e_1\}, \{e_2\}, \ldots, \{e_k\}, E(M)-(K\cup\{s_1, s_2\}))$$
is a path of $3$-separations in $M$ and, for all $i< k$, both $M/e_i$ and $M\backslash e_i$ have an $N$-minor. In particular, $e_i$ is $(N,B)$-robust for all $i<k$ and, consequently, $P$ is either $K\cup\{s_1, s_2\}$ or $K\cup\{s_1, s_2\}-\{e_k\}$. Thus, by Lemma~\ref{BS lem 6.1}, $(P,E(M)-P)$ is a sequential $3$-separation, completing the proof of (ii) and the theorem.
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
No House to Call My Home: Love, Family, and Other Transgressions
RYAN BERG with Melissa Febos
I first encountered Ryan Berg's writing as a jurist for a fellowship a few years back. Ryan's submission blew me away. It was an excerpt from his just-published debut, No House to Call My Home: Love, Family, and Other Transgressions, which details the lives of queer youths of color with whom he worked as a caseworker in a group home. Here was a work both political and narrative, compassionate and scrutinizing—I am always looking for such books. I had loved Adrian Nicole LeBlanc's Random Family and been so impressed by the power of marginalized people's stories to effect awareness and, hopefully, change. I already believed that narrative was a more accessible and powerful tool than rhetoric, and books like LeBlanc's and Berg's further proved this. In that 20-page sample I recognized what I have since read in the entirety of NHTCMH: the stories of queer homeless youths of color told with humility, self-scrutiny, intelligence, and love. It is a brave and conscientious book, an important book. Berg is mindful to keep his own story in the background of his subjects' and I was interested in hearing a bit more about his process.
Ryan Berg
(Nation Books, 2015)
Melissa Febos (Rail): Tell me the origin story of this book, Ryan.
Ryan Berg: I started working with LGBTQ youth in foster care in 2004. I'd been working in theater and was feeling the conversations within that community were taking place in silos. I felt a need to do something outside myself, so I applied to work at a group home. I hadn't intended to write about the youth, it felt too exploitive. I didn't want my writing to steer my experiences with them. During the summer of 2005 I went to the University of Iowa for a creative writing workshop for social workers and it was there I put down the first pages. I was encouraged by the instructor to continue writing after the workshop ended. Once I returned to New York I shelved the pages, and focused on working with the youth again. It wasn't until I left my position and started an MFA at Hunter College that I realized I couldn't shake these stories. I really felt an urgency in telling them. These stories felt unavoidable. There was nothing being said about the youth experience of homelessness, especially LGBTQ youth. We know the statistics. They're out there, but people really operate from empathy. I was trying to help by opening a door into an unseen world, to focus on the lives of the young people experiencing the hardships that were addressed in the book.
Rail: Our books are so profoundly different in about every superficial way, and many deeper ones, but I relate to this a lot. I also had an experience of explicitly thinking, I will not write about this. But the muse doesn't give a shit about our plans, does she? The story that needs you to write it comes calling, invited or not. I had already been hijacked by my memoir when I discovered the importance of writing it. To air that secret life had a value I hadn't ever considered before the impulse took me. But that story was mostly my own. Yours carried a different set of implications—can you talk about that?
Berg: There's a great responsibility in writing about others. The inherent challenge was how to do it while honoring their stories and their privacy. Writing about others—especially people already marginalized—creates such a skewed power dynamic. I wanted to make sure that I was exercising that power as justly as I could.
Rail: What does that mean to you, making sure that you're exercising the power as justly as you could?
Berg: I needed to interrogate my motives for writing the story. I was aware of my own privilege and power in that moment and tried not to shy away from it. Within the book I tried to acknowledge places where I can't comprehend or understand their experience, because I haven't been there. When they're experiencing family rejection or abuse or systemic racism, I acknowledge that, but I'm not really able to grasp all of that. It's about operating with as much empathy as possible.
Rail: I agree completely. I think that too often the people with privilege are unsure how to negotiate that position of power and choose to avoid acknowledging it, which is, of course, also a privilege. It's hard to know how to be an ally, and I think it's really one of the tasks of our generation to figure out that role—both how to ask for what we need, and to listen when others ask. I hope that as this conversation plays out more publicly, those in a position to bring attention to and acknowledge stories like those of the characters in your book choose to exercise the opportunity to acknowledge them, and offer this kind of empathic and nuanced illumination. How did the writing change over the course of working on the book?
Berg: I learned about humility writing this book, I learned about privilege. The writer who wrote the initial pages in 2005 was a much more sentimental, naive person. Over the course of writing the book I became more aware of issues of mass incarceration, systemic oppression, racial and economic justice, and the role of well-intentioned white social workers in the lives of young people of color. I learned about power, about the uncomfortable reality of my own power. I had to turn the lens inward and do some work on myself in order to attempt to do these stories justice. My hope is that this book creates space for LGBTQ youth experiencing homelessness to tell their own stories, that their own memoirs begin to emerge, becoming the necessary corrective to history.
Rail: Yeah, empirically I've learned that every story worth telling on the page is inevitably one that annihilates a part of the story I have about myself. Writing from a marginalized experience made me think (and in some cases made others ask) why me? Ultimately, I came to the same conclusion: we can't carry the burden of everyone's truth—no single book can—but every time we tell a story from that corner of humanity, we are making room for all the others that will come after ours. What do you think replaced the sentimentality?
Berg: I think what replaced it was knowing the ugly reality of our marginalized youth, and that we as a community are complicit in their neglect by remaining uneducated about the subject. There's no room for bleeding-heart sentimentality there. Because the stories, and the accompanying tragedies feel so overwhelming, the only way to write through it was to keep it simple, to boil it down to detail, to tell it clearly and in the moment. When I would try to talk about systemic failures and the enormity of the issues that lead to such dire outcomes, I would think, how do I even begin to comment on this?
Rail: Some reviewers seem to have been almost shocked by some of the unpleasant realities of the youths' lives. How do those kinds of reactions strike you?
Berg: It made me protective, because I grew to care a great deal for these young people, and I was tasked with helping them move forward. Yes, there's a great deal of tragedy, but there are moments of joy and connection. The unfortunate reality is that when you house numerous youth together who've all experienced great trauma in their lives, shit is going to get tense on a regular basis and eclipse the quiet moments. I find it strange that some people criticized the youth for acting out. It's true that the circumstances of these young people's lives are very disturbing, but all they experienced and expressed—the love, the loss, the turmoil, the betrayals—all these are universally human things.
Rail: You mentioned people being judgmental of the youth. How do you respond to those kinds of judgments?
Berg: I think that kind of judgment is a way to distance ourselves from the reality of poverty, homophobia, transphobia, and systemic oppression. If Bella's poverty is her fault, that means your financial comfort is solely your accomplishment, right? Nothing to do with the privileges afforded you. It's a very unsophisticated way of saying that if you work hard enough, anything is possible, everyone gets what they deserve. That's total bullshit. Many trans youth can't find legitimate work and face discrimination from service providers tasked with providing employment support. I work with homeless youth in Minneapolis now and witness some of them working forty hours a week, doing everything they've been told will help them break the cycle of poverty but they're never able to get a leg up. Politicians use this judgment of the individual as a tactic to distract us from a really fundamental question: Should we help people in need? The question shouldn't be who deserves help or not, but do we want to be a country that turns its back on the neediest. Vilification of individuals and their choices diverts us away from this question.
Rail: Victim-blaming is nearly as old a tradition as the myth of the American Dream, isn't it? I guess that's how we avoid having to look at the devastating social symptoms of our postcolonial legacy. Like, why can't all these historically subjugated people locate their damn bootstraps? There is no solution that doesn't threaten the very institutions our society is build upon—how much easier to tell them that it's their fault for not seizing the individualist spirit of this country. Do you feel the LGBTQ community has adequately demonstrated concern about the LGBTQ homeless youth problem?
Berg: Most of the major LGBTQ media outlets have been negligent in their reporting and dedication to this issue. I've read fifteen articles about Kim Davis this week, and one about LGBTQ homeless youth issues. Forty percent of homeless youth in the US identify as LGBTQ but make up only eight percent of the population. One in five transgender women report having experienced homelessness at one time or another in their lives. Many get turned away from shelter or services due to their gender identity. Where is the outrage in the community? I challenge the mainstream LGBTQ media to take an intentional stand on this issue , and dedicate themselves to reporting these stories. People often don't want to read about tragedy unless there is redemption in the end. Then we need to create that redemption. If we mobilized half as much and showed half the ingenuity as we did in the fight for marriage equality, the LGBTQ community, and their allies, could end LGBTQ youth homelessness. These young people have had to face enough indignities. They shouldn't have to face erasure and neglect from their own community.
Rail: How can the people reading this interview and your marvelous book be better allies to these kids, Ryan?
Berg: People need to educate themselves on the issue. I have a resource guide in the back of the book. I included this not only for youth to access services across the country but also for allies to reach out to service providers in their communities to find out how they can help. Policy is an important part of this issue. Don't wait for legislation. Politicians are reactive, they don't inform public opinion, they respond to it. Insist that LGBTQ best practices and cultural competency training is required for homeless youth workers and foster care parents so all queer-identified youth can feel affirmed and safe wherever they live. Aversion therapy needs to be banned, and LGBTQ-specific protections barring discrimination need to be in place. Clearly, there are many entry points. Calling your legislator to voice your concern in important; calling a youth shelter in your community to see if they are LGBTQ affirming in their practices is also a good place to start.
Melissa Febos
MELISSA FEBOS is the author of the memoir, Whip Smart. Her work has appeared in The Kenyon Review, Prairie Schooner, Guernica, Glamour, Post Road, Salon, New York Times, Dissent, Bitch Magazine, and elsewhere. Her essays have won prizes from Prairie Schooner, Story Quarterly, and The Center for Women Writers, and she is the recipient of fellowships from Bread Loaf, Virginia Center for Creative Arts, Vermont Studio Center, The Barbara Deming Memorial Fund, Lower Manhattan Cultural Council, and The MacDowell Colony. She is Assistant Professor of Creative Writing at Monmouth University and MFA faculty at the Institute of American Indian Arts (IAIA), and serves on the Executive Board of VIDA: Women in Literary Arts. www.melissafebos.com

redpajama_set_name: RedPajamaCommonCrawl


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BHOPAL, (GNI): Iconic Amitabh Bachchan at the inauguration of the first Kalyan Jewellers outlet on Raj Bhavan Road, Malviya Nagar with his wife Smt. Jaya Bachchan, Kalyan Jewellers Chairman and Managing Director T.S.Kalyanaraman and Executive Directors Rajesh Kalyanaraman and Ramesh Kalyanaraman.
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text:
//#define LOG_NDEBUG 0

#include <utils/Log.h>
#include <utils/RefBase.h>
#include <binder/Parcel.h>

#include <binder/IServiceManager.h>
#include <binder/IPCThreadState.h>

#include <display/MultiDisplayVideoClient.h>

namespace android {
namespace intel {

MultiDisplayVideoClient::MultiDisplayVideoClient() {
    mSessionId = -1;
    mVideo = NULL;
    mState = MDS_VIDEO_UNPREPARED;
};

MultiDisplayVideoClient::~MultiDisplayVideoClient() {
    // ALOGI("Destroy mds video client %d", mSessionId);
    close();
};

sp<IMDService> MultiDisplayVideoClient::getService() {
    sp<IServiceManager> sm = defaultServiceManager();
    if (sm == NULL) {
        ALOGW("%s: Failed to get service manager", __func__);
        return NULL;
    }
    sp<IMDService> mds = interface_cast<IMDService>(
            sm->getService(String16(INTEL_MDS_SERVICE_NAME)));
    if (mds == NULL) {
        ALOGW("%s: Failed to get MDS service", __func__);
        return NULL;
    }
    return mds;
}

void MultiDisplayVideoClient::close() {
    mSessionId = -1;
    mState = MDS_VIDEO_UNPREPARED;
    mVideo = NULL;
}

status_t MultiDisplayVideoClient::prepare(int state, bool isProtected) {
    if (state >= MDS_VIDEO_UNPREPARING && mSessionId < 0) {
        ALOGW("Invalid unprepared state");
        return UNKNOWN_ERROR;
    }
    if (state == mState) {
        // ALOGW("Same state %d", state);
        return UNKNOWN_ERROR;
    }
    if (!isProtected &&
            (state == MDS_VIDEO_PREPARING ||
             state == MDS_VIDEO_UNPREPARING)) {
        // ignore preparing and unpreparing state if video is not protected
        ALOGW("Ignore MDS preparing and unpreparing for clear content");
        return UNKNOWN_ERROR;
    }
    if (mVideo == NULL) {
        sp<IMDService> mds = getService();
        if (mds == NULL) {
            return UNKNOWN_ERROR;
        }
        mVideo = mds->getVideoControl();
    }
    if (mSessionId < 0) {
        mSessionId = mVideo->allocateVideoSessionId();
    }
    return NO_ERROR;
}

status_t MultiDisplayVideoClient::setVideoState(int state,
        bool isProtected, const sp<MetaData> &meta) {
    if (prepare(state, isProtected) != NO_ERROR)
        return UNKNOWN_ERROR;

    if (state == MDS_VIDEO_PREPARED) {
        MDSVideoSourceInfo info;
        memset(&info, 0, sizeof(MDSVideoSourceInfo));
        info.isProtected = isProtected;
        if (meta != NULL && meta.get() != NULL) {
            if (!meta->findInt32(kKeyFrameRate, &info.frameRate)) {
                info.frameRate = 0;
            }
            if (!meta->findInt32(kKeyDisplayWidth, &info.displayW)) {
                info.displayW = 0;
            }
            if (!meta->findInt32(kKeyDisplayHeight, &info.displayH)) {
                info.displayH = 0;
            }
        }
        mVideo->updateVideoSourceInfo(mSessionId, info);
        ALOGI("%s: Video source Info %d x %d @ %d fps", __func__,
               info.displayW, info.displayH, info.frameRate);
    }
    mVideo->updateVideoState(mSessionId, (MDS_VIDEO_STATE)state);
    if (state == MDS_VIDEO_UNPREPARED)
        close();
    mState = state;
    return NO_ERROR;
}

status_t MultiDisplayVideoClient::setVideoState(int state,
        bool isProtected, const sp<AMessage> &msg) {
    if (prepare(state, isProtected) != NO_ERROR)
        return UNKNOWN_ERROR;
    if (state == MDS_VIDEO_PREPARED) {
        if (msg != NULL && msg.get() != NULL) {
            MDSVideoSourceInfo info;
            memset(&info, 0, sizeof(MDSVideoSourceInfo));
            info.isProtected = isProtected;
            bool success = msg->findInt32("frame-rate", &info.frameRate);
            if (!success)
                info.frameRate = 0;
            success = msg->findInt32("width", &info.displayW);
            if (!success)
                info.displayW = 0;
            success = msg->findInt32("height", &info.displayH);
            if (!success)
                info.displayH = 0;
            mVideo->updateVideoSourceInfo(mSessionId, info);
            ALOGI("%s: Video source Info %d x %d @ %d fps ", __func__,
                     info.displayW, info.displayH, info.frameRate);
        }
    }
    mVideo->updateVideoState(mSessionId, (MDS_VIDEO_STATE)state);
    if (state == MDS_VIDEO_UNPREPARED)
        close();
    mState = state;
    return NO_ERROR;
}

status_t MultiDisplayVideoClient::reset() {
    sp<IMDService> mds = getService();
    if (mds == NULL) {
        return UNKNOWN_ERROR;
    }
    sp<IMultiDisplayVideoControl> video = mds->getVideoControl();
    if (video == NULL) {
        return UNKNOWN_ERROR;
    }
    return  video->resetVideoPlayback();
}

}; // namespace intel
}; // namespace android

redpajama_set_name: RedPajamaGithub


text:
Rineke Dijkstra (ur. 1959 w Sittard) – holenderska fotograficzka, artystka wideo, pracująca w Amsterdamie. Jej prace wystawiano m.in. w Tate Modern (2000), Muzeum Guggenheima w Nowym Jorku (2002, 2012) oraz na Biennale w Wenecji (2001).

Życiorys
Urodziła się w 1959 roku w Sittard. W latach 1981–1986 uczyła się fotografii w Gerrit Rietveld Academie w Amsterdamie. W 1984 w de Moor w Amsterdamie odbyła się jej pierwsza wystawa indywidualna, trzy lata później otrzymała nagrodę Kodak Award.

Przez kilka lat zajmowała się fotografią komercyjną, tworzoną na potrzeby biznesu. W 1990 roku artystka przeżyła poważny wypadek rowerowy, z powodu którego przez pięć miesięcy była unieruchomiona, a następnie musiała przejść długotrwałą rehabilitację. W 1991 roku Dijkstra wykonała autoportret, przedstawiający zmęczoną artystkę w kostiumie kąpielowym, tuż po wyczerpujących ćwiczeniach rehabilitacyjnych na basenie; fotografia ta stanowiła przełom w jej twórczości. Od 1992 Dijkstra pracowała nad serią zdjęć przedstawiających młodych ludzi w strojach kąpielowych pozujących m.in. na plażach amerykańskich, polskich, ukraińskich; cykl ten przyniósł artystce popularność.

Wystawy indywidualne artystki odbyły się m.in. w Galerie Nationale du Jeu de Paume w Paryżu (2005), Fundació la Caixa d'Estalvis Pensions w Barcelonie (2005), Rudolfinum w Pradze (2006), Marian Goodman Gallery w Nowym Jorku (2010), Tate Liverpool (2010). Jej prace pojawiły się na Biennale w Wenecji (1997, 2001), biennale w São Paulo (1998), Biennale Internationale di Fotografia w Turynie (1999) oraz na triennale fotografii i wideo w Nowym Jorku (2001).

W 2012 roku retrospektywy Dijkstry odbyły się w Muzeum Guggenheima w Nowym Jorku i w Muzeum Sztuki Współczesnej w San Francisco.

Twórczość
Artystka chętnie tworzy tematyczne serie fotografii. Często portretuje nastolatków, ludzi młodych znajdujących się u progu dojrzałości. Chętnie fotografuje też ludzi tuż po tym, jak podjęli duży fizyczny i emocjonalny wysiłek (np. kobiety po porodzie, toreadorzy po walce). Niektóre z jej fotografii to z kolei serie zdjęć tej samej osoby, dokumentujące jak zmieniała się ona na przestrzeni czasu.

Do jej fotografii należą m.in.:
 seria Beaches (1992–2002) przedstawiająca młodych ludzi na plażach, fotografowanych od przodu, w pozycji stojącej; zdjęcia były wykonywane m.in. na plażach USA, Polski, Ukrainy i Chorwacji,
 seria zdjęć portugalskich torreadorów fotografowanych tuż po zejściu z areny,
 seria Almerisa, na którą składają się zdjęcia Bośniaczki mieszkającej w holenderskim ośrodku dla uchodźców, wykonywane regularnie od 1994 przez ponad 10 lat,
 seria Oliver (2000-2003), na którą składają się zdjęcia młodego żołnierza Legii Cudzoziemskiej, dokumentujące jego wcielanie się w rolę legionisty,
 seria zdjęć izraelskich żołnierzy i żołnierek,
 seria Tiergarten (1998-2000), na którą składają się zdjęcia nastolatków wykonane w ogrodzie Großer Tiergarten.

Dijkstra jest też autorką sztuki wideo, do jej filmów należą m.in. I See A Woman Crying (Weeping Woman) (2009; film przedstawia grupę dzieci interpretujących obraz Picassa), Ruth Drawing Picasso (2009; film przedstawia dziewczynkę szkicującą obraz Picassa), The Krazyhouse (2009; film przedstawia grupę nastolatków tańczących do swojej ulubionej muzyki) i Annemiek (I wanna be with you) (1997).

Przypisy

Holenderscy artyści współcześni
Holenderscy fotografowie
Ludzie związani z Amsterdamem
Ludzie urodzeni w Sittard
Urodzeni w 1959

redpajama_set_name: RedPajamaWikipedia


Back in 2013, our family donned our running shoes to take on Everbank Field and had a blast. Three years later, with new bling, a new course and a new finish on the 50 yard line, little man and I were back in the game.
To be honest, I really didn't have any intention of running this. It's a tough race, incorporating lots of uphill ramps through the concourse and it's at night (I loathe running at night). Since I have practically zero 5Ks on my schedule and I didn't have other plans, I figured I would just go ahead and register. I knew that it would be a tough race, on the heels of a long run and a day out at a pool party, but how often does one get to run around a football stadium?
Little man has been a bit off running lately, so I asked him if he wanted to do the fun run. He was on the fence, and I decided that I could always just sign him up at the venue. The day of the race, he told me he wanted to do it, so I packed his sneakers and running clothes along with mine.
We arrived at the stadium at around 6:00 PM. There was a nasty storm hanging around the area, complete with black clouds and jags of lightning, so we were fairly certain registration would be easy. We were wrong. The line for day of registration was HUGE (in fact, we waited for about 35 minutes, before the race director pulled us and moved us to the front of the line so we could make the 6:45 fun run start – huge thanks to him for that!).
I grabbed little man's bib, and left Mr PugRunner to finish payment. I also left him with my bag, which had my phone, so I was hopeful I would be able to find him again before the start of my wave.
At 6:45, we were off. The fun run made a curve around the outside ground floor of the stadium, up a full set of ramps to the very top, and then wound through the concourse, going down some ramps, back up to the top, and then all the way down to wrap around to the entrance to the field. In order to reach the finish line, we had to go around most of the field, and crossed the line at 50 yard mark. So cool.
Little man was a beast. The last time he did this challenge, he managed to charm his way into a piggy back ride.
This time, he did it on his own, and he ran almost the whole way up the first flight of ramps. Rock star.
We were handed water and ushered up the stairs into the seating area, where Mr PugRunner was waiting for us. Little man collected his medal from one of the Jaguars cheerleaders and then we chilled out and watched more people cross the finish line, before it was time for me to head back to the start line.
I had signed up for the 7:45 wave – I wanted some of the experience of the lights on the field, without it being too dark and late. I toyed with the idea of switching to 8:15 to be with some friends, but I was also tired and wanted to go home. It made it tough to coordinate with friends, but I was able to hang out with S for a few.
At the start line, the race director outlined the course to us. He also mentioned that some of the latecomers didn't have timing chips on their bibs, so they could either keep track of their time via the race clock or their own personal timing devices, and then report it to the race team at the end. It was a nice gesture for those who wanted to be included in the official results.
We started promptly, and I felt very strong going out. I got to almost the one mile mark before starting my intervals – partly because the course was very tight at the start and partly because I felt good. By that point, we had taken care of the ground level part of the race, and were getting ready to enter the stadium.
While the weather was overcast and breezy, inside the stadium was nothing short of humid. The air was so thick you could chew it, and I looked forward to the brief moments of relief when we exited the concourse and got back out in the fresh air.
We had to climb two sets of ramps: the first set was a straight bottom-to-top go, while the second involved going up partway, around the concourse and then up the rest of the way.
It wasn't awful, but the downhill run was the best! I confess, I did stop at the very top of the stadium to snap this photo of the Hart Bridge (I'll see you in March!), but other than that, I kept on trucking.
Finally, we were back on the ground, through the field entrance, and I was making my loop of the football field. Mr PugRunner and little man spotted me from above and cheered me on – I loved being able to see them! A lot of people were slowing down to try to see (and get pictures of) themselves on the big screens, but I was more interested in getting across that finish line, so I just put my head down and did the work.
No PRs on this night, but I was ok with that. It was good to get out at night, good to run the shorter distance race and good to be outside my comfort zone.
rejoined the boys, and then went up the steps to the Bud Zone, where finishers could get their free beer or soda and some food.
I met up with a few friends, gave hugs and high fives, and then we were ready to go. It had been a long day and we were all ready for bed.
Despite its challenges, I'm glad I rallied for this one. It was good to stretch my legs, get in those inclines and push myself. And it was even better to run with my little guy. I've missed it.
Do you ever get to run in sports stadiums?
What's the most unique race venue you have experienced?
That is so cool! You are leading such a great example for him. Glad you all had so much fun!
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I am trying! I think if he's having fun, that's the main goal.
They had a Spartan down in the Tampa Bay Bucs Stadium and I remember people talking about all the stairs. Not so fun.
How fun to run a race at a football stadium! And at night! I have a hard time "feeling it" for a night race..so hard to get the nutrition thing down right. I always say this but your son always looks so proud with his medals! It's so cool that he loves doing this.
I had a hard time eating and drinking right for the day (it was especially hard since I was at a party and needed to drink water instead of having sangria), but once in a while it's ok. The sunset from the top of the stadium was worth it.
No, I never get to run in stadiums….and I wish I could!! This looks like a great challenge!
Those ramps would be the perfect blend of your running and stair climbing!
How much fun is running in a stadium to finish a race! Looks like little man loved it too! You really have some great photos too!
I did get to finish in the Manning Center which is a newly built football practice field at Ole Miss. We are Rebel fans so this was just awesome to me!
Oh, so cool! We are Miami Hurricanes, but since the old Orange Bowl was demolished, the only place we would have is the new Hard Rock Stadium (Hurricanes share with the Dolphins, which is my favorite pro team). Which I wouldn't mind, either.
Congrats to both of you! Sounds like a very fun event. The Soldier Field 10-Miler here finishes on the 50 yd line of the stadium. The most unique for me though might be the Indy 500 race track. That was so cool.
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We have the Daytona Half Marathon and it ends on the racetrack. I hear that it's a fabulous experience. One day I will give it a shot!
what a fun race!! little man looks amazing in his all his photos – congrats to him as well. I've run the Steelers 5K a few times. Love running in the stadium as a fan!
It's so fun to be in the stadium. To be honest, I would love to do this in Dolphins Stadium (Or Hard Rock Stadium, as I believe it's being called now), but maybe one day.
Since I live in CT, land of no sports teams, there are no races like this near me. It sounds AWESOME! I'm so glad you and Little Man both got to get out there. And even better that the Mr was able to come too! Sounds like a fabulous evening together!
LOL at land of no sports teams. Poor thing 🙁 We had a great night. Would have been better in the morning, but what can you do?
Awww congrats to your little man! he did great, he has grown quite a bit since his race, he looks really young in the piggy back pic!
Great job on the 5K – I have never run in a stadium but I can imagine it is challenging. I love the view from the top of the stadium.
Those ramps get pretty tough towards the end, especially in the humidity. But the view can't be beat. The beer was delicious.
What a fun and unique event!
I love the medal. Looks like an amazing event, I ran in a stadium once and loved it!
It was so much fun! And the medal was fabulous! They really upgraded them over the last few years!