Intersection Graphs with and without Product Structure

TL;DR

The paper investigates when intersection graphs formed from $\alpha$-free homothetic copies of a planar set $S$ admit product structure, revealing a threshold $\alpha^*(S)$ separating positive and negative results. It proves no product structure for many $S$ (including triangles and trapezoids) with $0<\alpha<1$, and shows for other $S$ (e.g., regular $n$-gons for infinitely many $n$) that $0<\alpha^*(S)<1$ with upper and lower bounds. For sufficiently large $\alpha$ in some polygon families, the induced graphs are planar and hence admit product structure, while a canonical drawing framework and $k$-independent crossing notion connect product structure to beyond-planar constraints. The nested grids construction provides a general method to certify nonexistence of product structure via lack of linear local treewidth. Overall, the work establishes a threshold-driven dichotomy for product structure in geometric intersection graphs and links planarity, canonical drawings, and crossing conditions to product-structure outcomes.

Abstract

A graph class $\mathcal{G}$ admits product structure if there exists a constant $k$ such that every $G \in \mathcal{G}$ is a subgraph of $H \boxtimes P$ for a path $P$ and some graph $H$ of treewidth $k$. Famously, the class of planar graphs, as well as many beyond-planar graph classes are known to admit product structure. However, we have only few tools to prove the absence of product structure, and hence know of only a few interesting examples of classes. Motivated by the transition between product structure and no product structure, we investigate subclasses of intersection graphs in the plane (e.g., disk intersection graphs) and present necessary and sufficient conditions for these to admit product structure. Specifically, for a set $S \subset \mathbb{R}^2$ (e.g., a disk) and a real number $α\in [0,1]$, we consider intersection graphs of $α$-free homothetic copies of $S$. That is, each vertex $v$ is a homothetic copy of $S$ of which at least an $α$-portion is not covered by other vertices, and there is an edge between $u$ and $v$ if and only if $u \cap v \neq \emptyset$. For $α= 1$ we have contact graphs, which are in most cases planar, and hence admit product structure. For $α= 0$ we have (among others) all complete graphs, and hence no product structure. In general, there is a threshold value $α^*(S) \in [0,1]$ such that $α$-free homothetic copies of $S$ admit product structure for all $α> α^*(S)$ and do not admit product structure for all $α< α^*(S)$. We show for a large family of sets $S$, including all triangles and all trapezoids, that it holds $α^*(S) = 1$, i.e., we have no product structure, except for the contact graphs (when $α= 1$). For other sets $S$, including regular $n$-gons for infinitely many values of $n$, we show that $0 < α^*(S) < 1$ by proving upper and lower bounds.

Figures (6)

• Figure 1: The strong product $P \boxtimes H$ with a row highlighted (left), a unit disk intersection representation of a graph $G$ (center), and its product structure representation $G \subseteq P \boxtimes H$ (right).
• Figure 2: $K_4$-free disk intersection graphs (left) and $K_3$-free unit ball intersection graphs (center) with no product structure, and a $\frac{1}{2}$-free disk intersection graph with the free area in blue (right).
• Figure 3: The $8$-gon segment with $4$ corners $\text{\Large \pentagon}_8^4$.
• Figure 4: Two regular $n$-gons meeting in third, $S$, for large $n$ and the cases $n \equiv 0, 1, 2, 3 \mod 4$.
• Figure 5: Values of $s(n)$ for $n = 3, \ldots, 100$.

Theorems & Definitions (8)

• Definition 1
• Definition 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 6
• Conjecture 7
• Theorem 8