A Freeable Matrix Characterization of Bipartite Graphs of Ferrers Dimension Three

TL;DR

The paper characterizes bipartite graphs of Ferrers dimension three, $\mathop{CHAIN}\nolimits^3$, as exactly those whose biadjacency matrices can be chosen to be free of the 3×3 patterns $\Gamma$ and $\Delta$, where $\Gamma$ and $\Delta$ are specific matrices defined with wildcard entries $*$ in $\mathbb{R}^3$. It provides a constructive equivalence: (i) any $G\in CHAIN^3$ admits an ordering yielding a $\Gamma$- and $\Delta$-free biadjacency representation, and (ii) any $\Gamma$- and $\Delta$-free biadjacency representation can be decomposed into an intersection of a CHAIN^2 graph and a CHAIN graph, establishing $G\in CHAIN^3$. The proof leverages a freeable-matrix framework with Hadamard products, and a detailed handling of submatrix patterns ($D=(1*01)$, $(1001)$, and $10^*$) to bridge geometric (3D orthants) and combinatorial (matrix-free) perspectives. This yields a practical, pattern-based criterion for recognizing and constructing CHAIN^3 graphs, with potential implications for related dimensional graph classes and their geometric representations.

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 $Γ= \begin{bmatrix} * & 1 & * \\ 1 & 0 & 1 \\ 0 & 1 & * \end{bmatrix}$ and $Δ= \begin{bmatrix} 1 & * & * \\ 0 & 1 & * \\ 1 & 0 & 1 \end{bmatrix}$, where $*$ denotes zero or one entry.

Figures (3)

• Figure 1: $A'$ cannot contain $D$ as a submatrix of $\Gamma$ or $\Delta$ on the left and right respectively.
• Figure 2: A construction of $A_{1,2}$ from matrix $A$. The ray-point representations of $G_1$ and $G_2$ are shown below and to the right of matrix $A$ respectively. The zero entries in the matrices are left blank. Notice that $A_{1,2}[4,1] \neq A[4,1]$ but other entries are equal.
• Figure 3: On the left, we have $j$ yet to be inserted into the order, and on the right, the order $L$ is depicted.

Theorems & Definitions (12)

• Theorem 1
• Lemma 1: das1989interval
• Lemma 2: hartman1991grid
• Lemma 3: shrestha2010orthogonal
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof