K-Circular Matroids of Graphs

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. We define the so-called {\em $k$-circular matroid} $M_k(G)$ on the edge set of graph $G$ for any non-negative integer $k$ so that $M(G) = M_0(G)$ and $M_{\theta }(G) = M_1(G)$. It is natural to consider the corresponding analog $(WP)_k$ of Whitney's problem $(WP)$ not only for $k=0$ and $k=1$ but also for any integer $k \ge 2$. In this paper we give a characterization of the $k$-circular matroid $M_k(G)$ by describing the main constituents (circuits, bases, and cocircuits) in terms of graph $G$ and establish some important properties of the $k$-circular matroid. The results of this paper will be used in our further research on the problem $(WP)_k$. In our next paper we use these results to study a particular problem of $(WP)_k$ on graphs uniquely defined by their $k$-circular matroids.