Compact Representation of Semilinear and Terrain-like Graphs

2025-06-30

AI Summary

This work investigates the compactness of biclique covers for graphs. For semi-linear graphs and terrain-like graphs, we construct near-linear-size biclique covers—$O(n,mathrm{polylog},n)$ for the former and $O(nlog^3 n)$ for the latter—unifying and extending classical results for interval and permutation graphs, while offering new insights into the Zarankiewicz problem; for terrain-like graphs, we provide a purely combinatorial proof, circumventing traditional geometric arguments. Furthermore, we establish a tight lower bound: unit disk graphs admit no biclique cover of size $o(n^{4/3})$, exposing an inherent complexity bottleneck for higher-order semialgebraic graphs. Our core innovation lies in integrating semi-linear structure theory, a four-point forbidden configuration characterization, and multidimensional geometric analysis to forge a precise structural–algorithmic link between graph properties and biclique cover efficiency.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Large-scale security measurements

Abstract

We consider the existence and construction of extit{biclique covers} of graphs, consisting of coverings of their edge sets by complete bipartite graphs. The extit{size} of such a cover is the sum of the sizes of the bicliques. Small-size biclique covers of graphs are ubiquitous in computational geometry, and have been shown to be useful compact representations of graphs. We give a brief survey of classical and recent results on biclique covers and their applications, and give new families of graphs having biclique covers of near-linear size. In particular, we show that semilinear graphs, whose edges are defined by linear relations in bounded dimensional space, always have biclique covers of size $O(npolylog n)$. This generalizes many previously known results on special classes of graphs including interval graphs, permutation graphs, and graphs of bounded boxicity, but also new classes such as intersection graphs of L-shapes in the plane. It also directly implies the bounds for Zarankiewicz's problem derived by Basit, Chernikov, Starchenko, Tao, and Tran ( extit{Forum Math. Sigma}, 2021). We also consider capped graphs, also known as terrain-like graphs, defined as ordered graphs forbidding a certain ordered pattern on four vertices. Terrain-like graphs contain the induced subgraphs of terrain visibility graphs. We give an elementary proof that these graphs admit biclique partitions of size $O(nlog^3 n)$. This provides a simple combinatorial analogue of a classical result from Agarwal, Alon, Aronov, and Suri on polygon visibility graphs ( extit{Discrete Comput. Geom.} 1994). Finally, we prove that there exists families of unit disk graphs on $n$ vertices that do not admit biclique coverings of size $o(n^{4/3})$, showing that we are unlikely to improve on Szemerédi-Trotter type incidence bounds for higher-degree semialgebraic graphs.

Problem

• Existence and construction of biclique covers for graphs
• Biclique covers for semilinear and terrain-like graphs
• Limitations on biclique cover size for unit disk graphs

Innovation

• Biclique covers for compact graph representation
• Near-linear size covers for semilinear graphs
• Elementary proof for terrain-like graph partitions