A Characterization of Circle Graphs in Terms of Multimatroid Representations

TL;DR

This paper characterizes circle graphs through multimatroid representations derived from isotropic matroids, establishing a new criterion based on field-representability of associated matroids.

Contribution

It provides a novel characterization of circle graphs using multimatroid representations linked to isotropic matroids, connecting graph theory with matroid theory over various fields.

Findings

G is a circle graph iff for every field, a representable matroid defines Z_3(G)

Connects multimatroid representations with existing circle graph characterizations

Establishes a field-independent criterion for circle graphs

Abstract

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.