Basis graphs of even Delta-matroids

A -matroid is a collection B of subsets of a finite set I, called bases, not necessarily equicardinal, satisfying the symmetric exchange property: For A,B ∈ B and i ∈ A B, there exists j ∈ B A such that (A {i, j}) ∈ B. A -matroid whose bases all have the same cardinality modulo 2 is called an even -matroid. The basis graph G=G(B) of an even -matroid B is the graph whose vertices are the bases of B and edges are the pairs A,B of bases differing by a single exchange (i.e., |A B| = 2). In this note, we present a characterization of basis graphs of even -matroids, extending the description of basis graphs of ordinary matroids given by S. Maurer in 1973: Theorem. A graph G= (V ,E) is a basis graph of an even -matroid if and only if it satisfies the following conditions: (a) if x1x2x3x4 is a square and b ∈ V , then d(b, x1)+ d(b, x3)= d(b, x2)+ d(b, x4); (b) each 2-interval of G contains a square and is an induced subgraph of the 4-octahedron; (c) the neighborhoods of vertices induce line graphs, or, equivalently, the neighborhoods of vertices do not contain induced 5and 6-wheels. (A 2-interval is the subgraph induced by two vertices at distance 2 and all their common neighbors; a square is an induced 4-cycle of G.) © 2006 Elsevier Inc. All rights reserved.