On a Class of Graphic Matrices

A \(\{0,1\}\)-matrix \(M\) is tree graphic if there exists a tree \(T\) such that the edges of \(T\) are indexed on the rows of \(M\) and the columns are the incidence vectors of the edge sets of paths of \(T\). Analogously, \(M\) is ditree graphic if there exists a ditree \(T\) such that the directed edges of \(T\) are indexed on the rows of \(M\) and the columns are the incidence vectors of the directed-edge sets of dipaths of \(T\). In this paper, a simple proof of an excluded-minor characterization of the class of tree-graphic matrices that are ditree-graphic is given. Then, using the same proof technique, a characterization of a “special” class of tree-graphic matrices (which are contained in the class of consecutive \(1\)’s matrices) is stated and proved.