A Freeable Matrix Characterization of Bipartite Graphs of Ferrers Dimension Three

TL;DR

This paper characterizes bipartite graphs of Ferrer dimension three using a specific matrix forbidden substructure, linking geometric intersection representations with matrix properties.

Contribution

It provides a novel matrix characterization of bipartite graphs with Ferrer dimension three, connecting geometric and algebraic graph representations.

Findings

Characterization of Ferrer dimension three graphs via forbidden matrices

Equivalence between geometric intersection and matrix conditions

Provides a matrix-based criterion for graph classification

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.