Contraction Bidimensionality of Geometric Intersection Graphs

Abstract

Given a graph G, we define \(\mathbf{bcg}(G)\) as the minimum k for which G can be contracted to the uniformly triangulated grid \(\Gamma _{k}\). A graph class \({\mathcal {G}}\) has the SQGC property if every graph \(G\in {\mathcal {G}}\) has treewidth \(\mathcal {O}(\mathbf{bcg}(G)^{c})\) for some \(1\le c<2\). The SQGC 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 SQGC property. This family includes, in particular, bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for contraction bidimensional problems.