Two matroidal families on the edge set of a graph (Q1613478)

Let \(X\) be a subset of vertices of a 2-connected undirected graph \(G\) with \(n\) vertices. The connected subgraphs of \(n\) edges whose unique circuit passes through at least one element of \(X\) are shown to be the bases of a matroid \(M(G)\). As opposed to the usual cycle matroid of \(G\), the matroid \(M(G)\) is not regular but representable over fields of sufficiently large cardinality. An amusing application dealing with the rigidity of some 2-dimensional bar-and-joint framework is given. It reminded the reviewer of the matroid of point splittings of a partial linear space; see Lemma 11 in \textit{M. Wild} [Discrete Math. 112, 207-244 (1993; Zbl 0808.06009)].