Two matroidal families on the edge set of a graph Original Research Article

Let G be a 2-connected undirected graph with n vertices. Its connected subgraphs of n−1 edges (that is, its spanning trees) are the bases of the usual cycle matroid of G. Let now X be a subset of vertices of G and consider those connected subgraphs of n edges whose unique circuit passes through at least one element of X. They are shown to be the bases of another matroid. A similar construction is given if the connectivity of the subgraph is not required but every circuit of the subgraph must pass through at least one element of X. Both constructions still lead to matroids if X is a subset of edges of G. Relation of the first construction to elementary strong maps (if G is planar) and representability properties of the matroids arising from these constructions are also presented. Finally, a civil engineering problem is described which served as the original motivation of this study.