A Freeable Matrix Characterization of Bipartite Graphs of Ferrers Dimension Three

2025-10-23

AI Summary

This paper addresses the problem of determining whether a bipartite graph has Ferrers dimension at most 3—that is, whether it can be realized as the intersection graph of axis-aligned boxes in ℝ³. The main contribution is the first forbidden submatrix characterization: a bipartite graph has Ferrers dimension ≤ 2 if and only if its biadjacency matrix avoids two specific 4×4 0–1 patterns—namely, the Γ-type and Δ-type configurations. Using combinatorial matrix analysis, structural decomposition of adjacency matrices, and explicit geometric constructions, the authors establish the necessity and sufficiency of this forbidden-pattern condition for the existence of a 3-dimensional Ferrers representation. This result provides an exact equivalence between a geometric intersection model and a combinatorial matrix property, completing the missing characterization for dimension three in Ferrers dimension theory. It further lays the foundational framework for higher-dimensional generalizations and algorithmic recognition.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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 $Gamma= egin{bmatrix} *&1&* \ 1&0&1 \ 0&1&* end{bmatrix}$ and $Delta = egin{bmatrix} 1&*&* \ 0&1&* \ 1&0&1 end{bmatrix}$, where $*$ denotes zero or one entry.

Problem

• Characterizing bipartite graphs with Ferrers dimension three
• Identifying forbidden matrix patterns for Ferrers dimension
• Establishing equivalence between geometric and matrix representations

Innovation

• Characterizes bipartite graphs via forbidden matrix patterns
• Links Ferrers dimension three to specific excluded submatrices
• Uses biadjacency matrix representation for graph classification