A minor-monotone graph parameter based on oriented matroids Original Research Article

For an undirected graph G = (V,E) let λ ′(G) be the largest d for which there exists an oriented matroid M on V of corank d such that for each nonzero vector (x+,x−) of M, x+ is nonempty and induces a connected subgraph of G. We show that λ′(G) is monotone under taking minors and clique sums. Moreover, we show that λ′(G) ⩽ 3 if and only if G has no K5- or V8-minor; that is, if and only if G arises from planar graphs by taking clique sums and subgraphs.