On the Boxicity of Line Graphs and of Their Complements

TL;DR

This paper introduces a new method for determining the boxicity of graphs using interval-order subgraphs, successfully applies it to complex graphs like Petersen and Kneser-graphs, and explores implications for line graphs and computational complexity.

Contribution

It presents a simple approach based on interval-order subgraphs for calculating boxicity, confirming conjectures, and enabling polynomial-time algorithms for certain graph problems.

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]. 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.