an:07158248
Zbl 1431.05032
Brijder, Robert; Traldi, Lorenzo
A characterization of circle graphs in terms of multimatroid representations
EN
Electron. J. Comb. 27, No. 1, Research Paper P1.25, 35 p. (2020).
00444109
2020
j
05B35 52B40
circle graph; multimatroid; delta-matroid; isotropic system; local equivalence; matroid; regularity; representation; unimodular orientation
Summary: The isotropic matroid \(M[IAS(G)]\) of a looped simple graph \(G\) is a binary matroid equivalent to the isotropic system of \(G\). In general, \(M[IAS(G)]\) is not regular, so it cannot be represented over fields of characteristic \(\neq 2\). The ground set of \(M[IAS(G)]\) is denoted \(W(G)\); it is partitioned into 3-element subsets corresponding to the vertices of \(G\). When the rank function of \(M[IAS(G)]\) is restricted to subtransversals of this partition, the resulting structure is a multimatroid denoted \(\mathcal{Z}_3(G)\). In this paper we prove that \(G\) is a circle graph if and only if for every field \(\mathbb{F} \), there is an \(\mathbb{F} \)-representable matroid with ground set \(W(G)\), which defines \(\mathcal{Z}_3(G)\) by restriction. We connect this characterization with several other circle graph characterizations that have appeared in the literature.