On graphs uniquely defined by their K-circular matroids

In 1930s Hassler Whitney considered and completely solved the problem (WP) of describing the classes of graphs G having the same cycle matroid M(G) (Whitney, 1933; Whitney, 1932). A natural analog (WP)' of Whitney's problem (WP) is to describe the classes of graphs G having the same matroid M'(G), where M'(G) is a matroid (on the edge set of G) distinct from M(G). For example, the corresponding problem (WP)' = (WP)(theta) for the so-called bicircular matroid M-theta(G) of graph G was solved in Coulard et al. (1991) and Wagner (1985). In De Jesus and Kelmans (2015) we introduced and studied the so-called k-circular matroids M-k(G) for every non-negative integer k that is a natural generalization of the cycle matroid M(G) := M-0(G) and of the bicircular matroid M-theta(G) := M-1(G) of graph G. In this paper (which is a continuation of our paper De Jesus and Kelmans (2015)) we establish some properties of graphs guaranteeing that the graphs are uniquely defined by their k-circular matroids. Published by Elsevier B.V.