On graphs uniquely defined by their $k$-circular matroids

In 30's Hassler Whitney considered and completely solved the problem $(WP)$ of describing the classes of graphs $G$ having the same cycle matroid $M(G)$. 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 by Coulard, Del Greco and Wagner. In our previous paper [arXive:1508.05364] 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 previous paper) we establish some properties of graphs guaranteeing that the graphs are uniquely defined by their $k$-circular matroids.