A splitter theorem for elastic elements in 3-connected matroids (Q6042103)

One of the most studied areas of matroid theory is the investigation of which elements may be removed from a matroid while retaining some level of connectivity, particularly \(3\)-connectivity. Key results include \textit{W. T. Tutte}'s wheels-and-whirls theorem [Can. J. Math. 18, 1301--1324 (1966; Zbl 0149.21501)] and \textit{P. D. Seymour}'s splitter theorem [J. Comb. Theory, Ser. B 28, 305--359 (1980; Zbl 0443.05027)], both of which provide crucial inductive tools for structural matroid theory. Another relevant result concerning \(3\)-connectivity is \textit{R. E. Bixby}'s lemma [Linear Algebra Appl. 45, 123--126 (1982; Zbl 0499.05020)] which states that in a \(3\)-connected matroid \(M\) with element \(e\), either the simplification of \(M/e\) or the cosimplification of \(M\setminus e\) is \(3\)-connected. The present paper studies elastic elements in \(3\)-connected matroids. An element \(e\) is elastic in a \(3\)-connected matroid \(M\) if both the simplification of \(M/e\) and the cosimplification of \(M\setminus e\) are \(3\)-connected. It builds on the work of \textit{G. Drummond} et al. [Electron. J. Comb. 28, No. 2, Research Paper P2.39, 15 p. (2021; Zbl 1466.05026)] where a wheels-and-whirls type theorem was established for elastic elements. Suppose that \(M\) is a \(3\)-connected matroid with \(3\)-connected minor \(N\). Then an element \(e\) is \(N\)-elastic if both the simplification of \(M/e\) and the cosimplification of \(M\setminus e\) are \(3\)-connected with an \(N\)-minor; an element is \(N\)-revealing if either the simplification of \(M/e\) has an \(N\)-minor but is not \(3\)-connected or the cosimplification of \(M\setminus e\) has an \(N\)-minor but is not \(3\)-connected. The main result of the paper says that if \(M\) has neither a \(4\)-element fan nor a second family of obstructions (known as \(\Theta\)-separators) whose elements reveal \(N\) in a particular way, then either \(M\) has at least two \(N\)-elastic elements, or whenever the simplication of \(M/e\) has an \(N\)-minor then it is \(3\)-connected and whenever the cosimplication of \(M\setminus e\) has an \(N\)-minor then it is \(3\)-connected. It is known from [Drummond et al., loc. cit.] that a \(3\)-connected matroid with no \(4\)-element fans and with no \(\Theta\)-separators has at least \(4\) elastic elements. In the current paper, the authors investigate the class of matroids for which this bound is tight and prove that any such matroid has pathwidth three. Finally, they show that certain earlier results, for example [\textit{J. Oxley} et al., Adv. Appl. Math. 41, No. 1, 1--9 (2008; Zbl 1139.05013)], concerned with removing an element relative to a fixed basis \(B\) (either contracting an element in \(B\) or deleting an element not in \(B\)) may be derived from their work.