A Freeable Matrix Characterization of Bipartite Graphs of Ferrers Dimension Three

Abstract

Ferrer dimension, along with the order dimension, is a standard dimensional concept for bipartite graphs. In this paper, we prove that a graph is of Ferrer dimension three (equivalent to the intersection bigraph of orthants and points in ${\mathbb R}^3$) if and only if it admits a biadjacency matrix representation that does not contain $\Gamma= \begin{bmatrix} * & 1 & * \\ 1 & 0 & 1 \\ 0 & 1 & * \end{bmatrix}$ and $\Delta = \begin{bmatrix} 1 & * & * \\ 0 & 1 & * \\ 1 & 0 & 1 \end{bmatrix}$, where $*$ denotes zero or one entry.