Isotropic matroids I: Multimatroids and neighborhoods

TL;DR

This paper explores properties of isotropic matroids derived from looped simple graphs, linking them to multimatroids and local graph equivalences, with applications to bipartite graphs and forests.

Contribution

It characterizes multimatroids associated with isotropic matroids and shows how isotropic matroids encode local graph equivalences and isomorphisms.

Findings

Characterization of multimatroids related to isotropic matroids

Graphs locally equivalent to bipartite graphs identified

Isomorphism of forests determined by isotropic matroids

Abstract

Several properties of the isotropic matroid of a looped simple graph are presented. Results include a characterization of the multimatroids that are associated with isotropic matroids and several ways in which the isotropic matroid of G incorporates information about graphs locally equivalent to G. Specific results of the latter type include a characterization of graphs that are locally equivalent to bipartite graphs, a direct proof that two forests are isomorphic if and only if their isotropic matroids are isomorphic, and a way to express local equivalence indirectly, using only edge pivots.