Detachable Pairs in 3-Connected Matroids

<p>The classical tool at the matroid theorist’s disposal when dealing with the common problem of wanting to remove a single element from a 3-connected matroid without losing 3-connectivity is Tutte’s Wheels-and-Whirls Theorem. However, situations arise where one wishes to delete or contract a pair of elements from a 3-connected matroid whilst maintaining 3-connectedness. The goal of this research was to provide a new tool for making such arguments. Let M be a 3-connected matroid. A detachable pair in M is a pair x, y ∈ E(M) such that either M\x, y or M/x, y is 3-connected. Naturally, our aim was to find the necessary conditions on M which guarantee the existence of a detachable pair. Triangles and triads are an obvious barrier to overcome, and can be done so by allowing the use of a Δ − Y exchange. Apart from these matroids with three-element 3-separating sets, the only other class of matroids that fail to contain a detachable pair for which no bound can be placed on the size of the ground set is the class of spikes. In particular, we prove the following result. Let M be a 3-connected matroid with at least thirteen elements. If M is not a spike, then either M contains a detachable pair, or there exists a matroid M′ where M′ is obtained by performing a single Δ − Y exchange on either M or M* such that M′ contains a detachable pair. As well as being an important theorem in its own right, we anticipate that this result will be essential in future attempts to extend Seymour’s Splitter Theorem in a comparable manner; where the goal would be to obtain a detachable pair as well as maintaining a 3-connected minor. As such, much work has been done herein to study the precise configurations that arise in 3-separating subsets which themselves yield no detachable pair.</p>

Detachable Pairs in 3-Connected Matroids

<p>The classical tool at the matroid theorist’s disposal when dealing with the common problem of wanting to remove a single element from a 3-connected matroid without losing 3-connectivity is Tutte’s Wheels-and-Whirls Theorem. However, situations arise where one wishes to delete or contract a pair of elements from a 3-connected matroid whilst maintaining 3-connectedness. The goal of this research was to provide a new tool for making such arguments. Let M be a 3-connected matroid. A detachable pair in M is a pair x, y ∈ E(M) such that either M\x, y or M/x, y is 3-connected. Naturally, our aim was to find the necessary conditions on M which guarantee the existence of a detachable pair. Triangles and triads are an obvious barrier to overcome, and can be done so by allowing the use of a Δ − Y exchange. Apart from these matroids with three-element 3-separating sets, the only other class of matroids that fail to contain a detachable pair for which no bound can be placed on the size of the ground set is the class of spikes. In particular, we prove the following result. Let M be a 3-connected matroid with at least thirteen elements. If M is not a spike, then either M contains a detachable pair, or there exists a matroid M′ where M′ is obtained by performing a single Δ − Y exchange on either M or M* such that M′ contains a detachable pair. As well as being an important theorem in its own right, we anticipate that this result will be essential in future attempts to extend Seymour’s Splitter Theorem in a comparable manner; where the goal would be to obtain a detachable pair as well as maintaining a 3-connected minor. As such, much work has been done herein to study the precise configurations that arise in 3-separating subsets which themselves yield no detachable pair.</p>

A Splitter Theorem for 3-Connected 2-Polymatroids

Seymour's Splitter Theorem is a basic inductive tool for dealing with $3$-connected matroids. This paper proves a generalization of that theorem for the class of $2$-polymatroids. Such structures include matroids, and they model both sets of points and lines in a projective space and sets of edges in a graph. A series compression in such a structure is an analogue of contracting an edge of a graph that is in a series pair. A $2$-polymatroid $N$ is an s-minor of a $2$-polymatroid $M$ if $N$ can be obtained from $M$ by a sequence of contractions, series compressions, and dual-contractions, where the last are modified deletions. The main result proves that if $M$ and $N$ are $3$-connected $2$-polymatroids such that $N$ is an s-minor of $M$, then $M$ has a $3$-connected s-minor $M'$ that has an s-minor isomorphic to $N$ and has $|E(M)| - 1$ elements unless $M$ is a whirl or the cycle matroid of a wheel. In the exceptional case, such an $M'$ can be found with $|E(M)| - 2$ elements.