Bidimensionality, Map Graphs, and Grid Minors

TL;DR

This paper extends bidimensionality theory to map and power graphs, establishing polynomial relations between treewidth and grid minors, which improves fixed-parameter algorithm efficiencies.

Contribution

It introduces a novel technique linking treewidth and grid minors for non-minor-closed graph families, enhancing algorithmic bounds.

Findings

Polynomial relation between treewidth and grid minors for map and power graphs

Improved fixed-parameter algorithm running times

Linear relation between treewidth of bounded-genus graphs and their duals

Abstract

In this paper we extend the theory of bidimensionality to two families of graphs that do not exclude fixed minors: map graphs and power graphs. In both cases we prove a polynomial relation between the treewidth of a graph in the family and the size of the largest grid minor. These bounds improve the running times of a broad class of fixed-parameter algorithms. Our novel technique of using approximate max-min relations between treewidth and size of grid minors is powerful, and we show how it can also be used, e.g., to prove a linear relation between the treewidth of a bounded-genus graph and the treewidth of its dual.