On the Boxicity of Line Graphs and of Their Complements

2025-01-09

AI Summary

This study investigates the boxicity—the minimum dimension $ d $ such that a graph is representable as an intersection graph of axis-aligned hyperrectangles in $ mathbb{R}^d $—of line graphs and their complements, a problem known to be NP-hard. We introduce a general analytical framework based on interval-order subgraphs to systematically characterize structural properties and extremal behavior of boxicity. Our contributions include: (i) the first exact determination that the Petersen graph has boxicity 3; (ii) a proof of the conjecture that the Kneser graph $ K(n,2) $ has boxicity $ n-2 $ for all $ n geq 5 $; (iii) the discovery that any line graph admits only $ O(m^k) $ edge-maximal interval-order subgraphs, where $ m $ is the number of edges—thereby extending the polynomial-time solvability frontier for this NP-hard problem; and (iv) the derivation of polynomial-time algorithms for deciding the existence of an interval-order subgraph, computing an optimal one, and—under bounded boxicity—computing both the boxicity and the interval completion of the complement of a line graph.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The boxicity of a graph is the smallest dimension $d$ allowing a representation of it as the intersection graph of a set of $d$-dimensional axis-parallel boxes. We present a simple general approach to determining the boxicity of a graph based on studying its ``interval-order subgraphs.'' The power of the method is first tested on the boxicity of some popular graphs that have resisted previous attempts: the boxicity of the Petersen graph is $3$, and more generally, that of the Kneser-graphs $K(n,2)$ is $n-2$ if $nge 5$, confirming a conjecture of Caoduro and Lichev [Discrete Mathematics, Vol. 346, 5, 2023]. As every line graph is an induced subgraph of the complement of $K(n,2)$, the developed tools show furthermore that line graphs have only a polynomial number of edge-maximal interval-order subgraphs. This opens the way to polynomial-time algorithms for problems that are in general $NP$-hard: for the existence and optimization of interval-order subgraphs of line graphs, or of interval completions and the boxicity of their complement, if the boxicity is bounded. We finally extend our approach to upper and lower bounding the boxicity of line graphs.

Problem

• boxicity
• interval graphs
• polynomial time algorithms

Innovation

• Interval-order Subgraphs
• Boxicity Determination
• Polynomial-time Algorithms