A Characterization of Circle Graphs in Terms of Multimatroid Representations

Abstract

A graph G has an associated multimatroid Z3(G), which is equivalent to the isotropic system of G studied by Bouchet. In previous work it was shown that G is a circle graph if and only if for every field F, the rank function of Z3(G) can be extended to the rank function of an Frepresentable matroid. In the present paper we strengthen this result using a multimatroid analogue of total unimodularity. As a consequence we obtain a characterization of matroid planarity in terms of this totalunimodularity analogue.Definition 4. If G is a looped simple graph then Z 3 (G) is the 3-matroid (W (G), Ω, r), where Ω is the set of vertex triples of vertices of G and r is given by the GF (2)-rank of sets of columns of IAS(G). Also,The multimatroids Z 2 (G) and Z 3 (G) are equivalent to the delta-matroid and isotropic system of G, respectively. That is, if G and H are looped simple graphs then Z 2 (G) and Z 2 (H) are isomorphic 2-matroids if and only if the deltamatroids of G and H are isomorphic; and Z 3 (G) and Z 3 (H) are isomorphic 3matroids if and only if the isotropic systems of G and H are isomorphic. Despite