Online Talk: George Drummond

YouTube recording:

Time: Thursday, July 6, 3pm ET

Speaker: George Drummond, University of Canterbury
Title: Elastic elements and where to find them

Abstract: In this talk we will discuss the presence of “elastic” elements in 3-connected matroids. An element e of a 3-connected matroid M is elastic if both the simplification of M/e and the cosimplification of M\e are 3-connected. Those familiar with Bixby’s Lemma will recall that at least one of these two conditions holds for every element. We will consider the obstructions to the presence of elastic elements, namely 4-element fans and another family of 3-separating structures we dub “theta-separators”, before arriving at our wheels-and-whirls-type theorem for elastic elements. In particular, it is shown that if neither of the aforementioned obstructions is present, then we are guaranteed at least four elastic elements. This wheels-and-whirls-type result then extends naturally to a splitter theorem, where the removal of elements is with respect to elasticity and keeping a specified 3-connected minor. We will also consider some applications of these results to the related study of removing elements relative to a fixed basis.

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