Contraction Bidimensionality of Geometric Intersection Graphs

TL;DR

This paper introduces the SQG${\bf C}$ property for graph classes, linking contraction measures to treewidth, and demonstrates that many classes, including bounded-degree string graphs, satisfy this property, broadening the scope of bidimensionality theory.

Contribution

The paper establishes that a broad family of graph classes, including bounded-degree string graphs, satisfy the SQG${\bf C}$ property, extending bidimensionality theory applications.

Findings

Bounded-degree string graphs satisfy the SQG${\bf C}$ property.

The SQG${\bf C}$ property links contraction measures to treewidth.

This extension broadens the applicability of bidimensionality theory.

Abstract

Given a graph $G$, we define ${\bf bcg}(G)$ as the minimum $k$ for which $G$ can be contracted to the uniformly triangulated grid $\Gamma_{k}$. A graph class ${\cal G}$ has the SQG${\bf C}$ property if every graph $G\in{\cal G}$ has treewidth $\mathcal{O}({\bf bcg}(G)^{c})$ for some $1\leq c<2$. The SQG${\bf C}$ property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a wide family of graph classes that satisfy the SQG${\bf C}$ property. This family includes, in particular, bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for contraction bidimensional problems.