Elastic Elements in $3$-Connected Matroids

A result widely used in the study of $3$-connected matroids is due to Bixby [1]: if $e$ is an element of a $3$-connected matroid $M$, then either $M\backslash e$ or $M/e$ has no non-minimal $2$-separations, in which case, ${\rm co}(M\backslash e)$, the cosimplification of $M$, or ${\rm si}(M/e)$, the simplification of $M$, is $3$-connected. A $2$-separation $(X,Y)$ is minimal if $\min\{|X|,|Y|\}=2$. This result is commonly referred to as Bixby’s Lemma. Thus, although an element $e$ of a $3$-connected matroid $M$ may have the property that neither $M\backslash e$ nor $M/e$ is $3$-connected, Bixby’s Lemma says that at least one of $M\backslash e$ and $M/e$ is close to being $3$-connected in a very natural way. In this paper, we are interested in whether or not there are elements $e$ in $M$ such that both ${\rm co}(M\backslash e)$ and ${\rm si}(M/e)$ are $3$-connected, in which case, we say $e$ is elastic. In general, a $3$-connected matroid $M$ need not have any elastic elements. For example, all wheels and whirls of rank at least four have no elastic elements. The reason for this is that every element of such a matroid is in a $4$-element fan and, geometrically, every $4$-element fan is positioned in a certain way relative to the rest of the elements of the matroid. However, $4$-element fans are not the only obstacles to $M$ having elastic elements.

Let $n\geqslant 3$, and let $Z=\{z_{1},z_{2},\ldots,z_{n}\}$ be a basis of $PG(n-1,\mathbb{R})$. Suppose that $L$ is a line that is freely placed relative to $Z$. For each $i\in\{1,2,\ldots,n\}$, let $w_{i}$ be the unique point of $L$ contained in the hyperplane spanned by $Z-\{z_{i}\}$. Let $W=\{w_{1},w_{2},\ldots,w_{n}\}$, and let $\Theta_{n}$ denote the restriction of $PG(n-1,\mathbb{R})$ to $W\cup Z$. Note that $\Theta_{n}$ is $3$-connected and $Z$ is a corank-$2$ subset of $\Theta_{n}$. For all $i\in\{1,2,\ldots,n\}$, we denote the matroid $\Theta_{n}\backslash w_{i}$ by $\Theta^{-}_{n}$. The matroid $\Theta^{-}_{n}$ is well defined as, up to isomorphism, $\Theta_{n}\backslash w_{i}\cong\Theta_{n}\backslash w_{j}$ for all $i,j\in\{1,2,\ldots,n\}$. For the interested reader, the matroid $\Theta_{n}$ underlies the matroid operation of segment-cosegment exchange [6] which generalises the operation of delta-wye exchange. A more formal definition of $\Theta_{n}$ is given in Section 5.

If $n=3$, then $\Theta_{3}$ is isomorphic to $M(K_{4})$. However, for all $n\geqslant 4$, the matroid $\Theta_{n}$ has no $4$-element fans and, also, no elastic elements. Furthermore, for all $n\geqslant 3$, the set $W$ is a modular flat of $\Theta_{n}$ [6]. Thus, if $M$ is a matroid and $W$ is a subset of $E(M)$ such that $M|W\cong U_{2,n}$, then the generalised parallel connection $P_{W}(\Theta_{n},M)$ of $\Theta_{n}$ and $M$ exists. In particular, it is straightforward to construct $3$-connected matroids having no $4$-element fans and no elastic elements. For example, take $U_{2,n}$ and repeatedly use the generalised parallel connection to “attach” copies of $\Theta_{k}$, where $4\leqslant k\leqslant n$, to any $k$-element subset of the elements of $U_{2,n}$.

Let $M$ be a $3$-connected matroid, and let $A$ and $B$ be rank-$2$ and corank-$2$ subsets of $E(M)$. We say that $A\cup B$ is a $\Theta$-separator of $M$ if $r(M)\geqslant 4$ and $r^{*}(M)\geqslant 4$, and either $M|(A\cup B)$ or $M^{*}|(A\cup B)$ is isomorphic to one of the matroids $\Theta_{n}$ and $\Theta^{-}_{n}$ for some $n\geqslant 3$. We will show in Section 5 that if $S$ is a $\Theta$-separator of $M$, then $S$ contains at most one elastic element. Note that if $r(M)=3$, then ${\rm si}(M/e)$ is $3$-connected for all $e\in E(M)$, while if $r^{*}(M)=3$, then ${\rm co}(M\backslash e)$ is $3$-connected for all $e\in E(M)$. The main theorem of this paper is that, alongside $4$-element fans, $\Theta$-separators are the only obstacles to elastic elements in $3$-connected matroids.

A $3$-separation $(A,B)$ of a matroid is vertical if $\min\{r(A),r(B)\}\geqslant 3$. Now, let $M$ be a matroid and let $(X,\{e\},Y)$ be a partition of $E(M)$. We say that $(X,\{e\},Y)$ is a vertical $3$-separation of $M$ if $(X\cup\{e\},Y)$ and $(X,Y\cup\{e\})$ are both vertical $3$-separations and $e\in{\rm cl}(X)\cap{\rm cl}(Y)$. Furthermore, $Y\cup\{e\}$ is maximal in this separation if there exists no vertical $3$-separation $(X^{\prime},\{e^{\prime}\},Y^{\prime})$ of $M$ such that $Y\cup\{e\}$ is a proper subset of $Y^{\prime}\cup\{e^{\prime}\}$. Essentially, all of the work in the paper goes into establishing the following theorem.

Note that, in the context of Theorem 1, if $X\cup\{e\}$ is a $4$-element fan, then it is possible that $X$ contains two elastic elements. For example, consider the rank-$4$ matroids $M_{1}$ and $M_{2}$ for which geometric representations are shown in Fig. 1. For each ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i}\in\{1,2\}$, the tuple $F=(e_{1},e_{2},e_{3},e_{4})$ is a $4$-element fan of $M_{i}$ and $(F-\{e_{1}\},\{e_{1}\},E(M_{i})-F)$ is a vertical $3$-separation of $M_{i}$. In $M_{1}$, none of $e_{2}$, $e_{3}$, and $e_{4}$ are elastic, while in $M_{2}$, both $e_{2}$ and $e_{3}$ are elastic. However, provided $X\cup\{e\}$ is a maximal fan, the instance illustrated in Fig. 1(i) is essentially the only way in which $X$ does not contain two elastic elements. This is made more precise in Section 3. As noted above, if $X$ is contained in a $\Theta$-separator, then $X$ contains at most one elastic element. The details of the way in which this happens is given in Section 5.

An almost immediate consequence of Theorem 1 is the following corollary.

The condition in Corollary 2 that $M$ has no $4$-element fans and no $\Theta$-separators is not necessarily that restrictive. For example, if $N$ is an excluded minor for $GF(q)$-representability (or, more generally, for $\mathbb{P}$-representability, where $\mathbb{P}$ is a partial field), then $N$ has no $4$-element fans and no $\Theta$-separators. The fact that $N$ has no $4$-element fans is well known and straightforward to show. To see that $N$ has no $\Theta$-separators, suppose that $N$ has a $\Theta$-separator. By duality, we may assume that $N$ has rank-$2$ and corank-$2$ sets $W$ and $Z$, respectively, such that $M|(W\cup Z)$ is isomorphic to either $\Theta_{n}$ or $\Theta^{-}_{n}$, for some $n\geqslant 3$. Say $M|(W\cup Z)$ is isomorphic to $\Theta_{n}$. Then the matroid $N^{\prime}$ obtained from $N$ by a cosegment-segment exchange on $Z$ is isomorphic to the matroid obtained from $N$ by deleting $Z$ and, for each $w\in W$, adding an element in parallel to $w$. It is shown in [6, Theorem 1.1] that the class of excluded minors for $GF(q)$-representability (or, more generally, $\mathbb{P}$-representability) is closed under the operation of cosegment-segment exchange, and so $N^{\prime}$ is also an excluded minor for $GF(q)$-representability. But $N^{\prime}$ contains elements in parallel, a contradiction. The same argument holds if $M|(W\cup Z)$ is isomorphic to $\Theta^{-}_{n}$ except that, in applying a cosegment-segment exchange, we additionally add an element freely in the span of $W$.

Like Bixby’s Lemma, Corollary 2 is an inductive tool for handling the removal of elements of $3$-connected matroids while preserving connectivity. The most well-known examples of such tools are Tutte’s Wheels-and-Whirls Theorem [9] and Seymour’s Splitter Theorem [8]. In both theorems, this removal preserves $3$-connectivity. More recently, there have been analogues of these theorems in which the removal of elements preserves $3$-connectivity up to simplification and cosimplification. These analogues have additional conditions on the elements being removed. Let $B$ be a basis of a $3$-connected matroid $M$, and suppose that $M$ has no $4$-element fans. Say $M$ is representable over some field $\mathbb{F}$ and that we are given a standard representation of $M$ over $\mathbb{F}$. To keep the information displayed by the representation in an $\mathbb{F}$-representation of a single-element deletion or a single element contraction of $M$, we need to avoid pivoting. To do this, we want to either contract an element in $B$ or delete an element in $E(M)-B$. Whittle and Williams [11] showed that if $|E(M)|\geqslant 4$, then $M$ has at least four elements $e$ such that either ${\rm si}(M/e)$ is $3$-connected if $e\in B$ or ${\rm co}(M\backslash e)$ is $3$-connected if $e\in E(M)-B$. Brettell and Semple [2] establish a Splitter Theorem counterpart to this last result where, again, $3$-connectivity is preserved up to simplification and cosimplification. These last two results are related to an earlier result of Oxley et al. [5]. Indeed, the starting point for the proof of Theorem 1 is [5].

The paper is organised as follows. The next section contains some necessary preliminaries on connectivity, while Section 3 considers fans and determines exactly which elements of a fan are elastic. Section 4 establishes two results concerning when an element in a rank-$2$ restriction of a $3$-connected matroid is deletable or contractible, and Section 5 considers $\Theta$-separators, and determines the elasticity of the elements of these sets. Section 6 consists of the proofs of Theorem 1 and Corollary 2. Effectively, all of the work that proves these two results goes into proving Theorem 1. We break the proof of Theorem 1 into two lemmas depending on whether or not $X$ contains at least one element that is not contractible. The statements of these lemmas, Lemma 17 and Lemma 18, provide additional structural information when $X$ is contained in a $\Theta$-separator. Throughout the paper, the notation and terminology follows [3].

Let $M$ be a $3$-connected matroid. A subset $F$ of $E(M)$ with at least three elements is a fan if there is an ordering $(f_{1},f_{2},\ldots,f_{k})$ of $F$ such that

1. (i)

   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

2. (ii)

   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.

If $k\geqslant 4$, then the elements $f_{1}$ and $f_{k}$ are the ends of $F$. Furthermore, if $\{f_{1},f_{2},f_{3}\}$ is a triangle, then $f_{1}$ is a spoke-end; otherwise, $f_{1}$ is a rim-end. Observe that if $F$ is a $4$-element fan $(f_{1},f_{2},f_{3},f_{4})$, then either $f_{1}$ or $f_{4}$ is the unique spoke-end of $F$ depending on whether $\{f_{1},f_{2},f_{3}\}$ or $\{f_{2},f_{3},f_{4}\}$ is a triangle, respectively. The proof of the next lemma is straightforward and omitted.

We end this section by determining when an element in a fan of size at least four is elastic. For subsets $X$ and $Y$ of a matroid, the local connectivity between $X$ and $Y$, denoted $\sqcap(X,Y)$, is defined by

Let $M$ be a $3$-connected matroid and let $k$ be a positive integer. A flower $\Phi$ of $M$ is an (ordered) partition $(P_{1},P_{2},\ldots,P_{k})$ of $E(M)$ such that each $P_{i}$ has at least two elements and is $3$-separating, and each $P_{i}\cup P_{i+1}$ is $3$-separating, where all subscripts are interpreted modulo $k$. If $k\geqslant 4$, we say $\Phi$ is swirl-like if $\bigcup_{i\in I}P_{i}$ is exactly $3$-separating for all proper subsets $I$ of $\{1,2,\ldots,k\}$ whose members form a consecutive set in the cyclic order $(1,2,\ldots,k)$, and

for all distinct $i,j\in\{1,2,\ldots,k\}$. For further details of swirl-like flowers and, more generally flowers, we refer the reader to [4].

Let $M$ be a matroid. A subset $L$ of $E(M)$ of size at least two is a segment if $M|L$ is isomorphic to a rank-$2$ uniform matroid. In this section we consider when an element in a segment is deletable or contractible. We begin with the following elementary lemma.