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)θ for the so-called bicircular matroid Mθ(G) of graph G was solved in Coulard et al. (1991) and Wagner (1985). In De Jesús and Kelmans (2015) we introduced and studied the so-called k-circular matroids Mk(G) for every non-negative integer k that is a natural generalization of the cycle matroid M(G):=M0(G) and of the bicircular matroid Mθ(G):=M1(G) of graph G. In this paper (which is a continuation of our paper De Jesús and Kelmans (2015)) we establish some properties of graphs guaranteeing that the graphs are uniquely defined by their k-circular matroids.