Boxicity and Interval-Orders: Petersen and the Complements of Line Graphs

TL;DR

This paper introduces a new method for determining graph boxicity using interval-order subgraphs, successfully applied to Petersen and Kneser graphs, and explores implications for line graphs and related computational problems.

Contribution

The paper presents a simple approach to compute graph boxicity via interval-order subgraphs and applies it to solve longstanding open problems for Petersen and Kneser graphs.

Findings

Boxicity of Petersen graph is 3.

Boxicity of Kneser-graphs K(n,2) is n-2 for n ≥ 5.

Line graphs have polynomially many edge-maximal interval-order subgraphs.

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 $n\ge 5$, confirming a conjecture of Caoduro and Lichev [Discrete Mathematics, Vol. 346, 5, 2023]. Since 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 $\mathcal{NP}$-hard: for the existence and optimization of interval-order subgraphs of line-graphs, or of interval-completions of their complement.