A Characterization of Circle Graphs in Terms of Multimatroid Representations

  • Robert Brijder
  • Lorenzo Traldi
Keywords: circle graph, multimatroid, delta-matroid, isotropic system, local equivalence, matroid, regularity, representation, unimodular orientation


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.
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