On the Boxicity of Line Graphs and of Their Complements

TL;DR

This work develops a general interval-order framework to bound and compute the boxicity of line graphs and their complements. By analyzing maximal interval-order subgraphs, it derives a precise boxicity for complements of line graphs, notably proving $\mathrm{box}(\overline{L(K_n)})=n-2$ for $n\ge5$ and establishing polynomial-time interval completions, enabling an XP algorithm for boxicity decisions on these graphs. It also provides new upper and lower bounds for the boxicity of line graphs themselves, including an upper bound of $\lceil 5\log n\rceil$ and a nontrivial logarithmic lower bound for $L(K_n)$. The results yield polynomial-time methods for related problems (e.g., Interval Graph Completion) on complements of line graphs and extend the toolbox for studying Kneser graphs and the Petersen graph within the boxicity landscape.

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.

Figures (4)

• Figure 1: A graph $G$ with an ordering $\sigma : = (v_1, \ldots, v_5)$ of $V(G)$, the corresponding $G_i$ for $i \in [3]$, and $G^\sigma$. For each $G_i$ ($i \in [3]$), the vertices of $V_i$ are marked with red disks. Note that $v_4$, the only vertex of $V_3$, is not contained in $N_G(v_4)$, so $E_4 = \emptyset$.
• Figure 2: The edges of $V(K_n)$ that can be contained in $\sigma$ in Case 1. Dashed red lines represent the vertices in $N^+ (v_3)$.
• Figure 3: The three edges of $V(K_n)$ that $\sigma$ may start within Case 2.
• Figure 4: An illustration of Case 3 and its two subcases.

Theorems & Definitions (35)

• theorem 1
• lemma 1
• theorem 2
• theorem 3
• theorem 4
• theorem 5
• lemma 2: Roberts 1969_Roberts, 1969
• lemma 3: Cozzens and Roberts 1983_Cozzens, 1983
• lemma 4: Roberts 1969_Roberts, 1969
• theorem 6: Roberts 1969_Roberts, 1969