Recognizing Map Graphs of Bounded Treewidth

TL;DR

This paper introduces a fixed-parameter tractable algorithm for recognizing map graphs with bounded treewidth, providing efficient recognition and certificates, and extends to testing for k-maps and hole-free k-maps.

Contribution

It presents the first linear-time fixed-parameter algorithm for recognizing map graphs parameterized by treewidth, within a broader framework for k-maps and hole-free k-maps.

Findings

Algorithm is linear in graph size for recognizing map graphs.

Recognition is fixed-parameter tractable with respect to treewidth.

Extends to testing for k-maps and hole-free k-maps.

Abstract

A map graph is a graph admitting a representation in which vertices are nations on a spherical map and edges are shared curve segments or points between nations. We present an explicit fixed-parameter tractable algorithm for recognizing map graphs parameterized by treewidth. The algorithm has time complexity that is linear in the size of the graph and, if the input is a yes-instance, it reports a certificate in the form of a so-called witness. Furthermore, this result is developed within a more general algorithmic framework that allows to test, for any $k$, if the input graph admits a $k$-map (where at most $k$ nations meet at a common point) or a hole-free~$k$-map (where each point of the sphere is covered by at least one nation). We point out that, although bounding the treewidth of the input graph also bounds the size of its largest clique, the latter alone does not seem to be a strong enough structural limitation to obtain an efficient time complexity. In fact, while the largest clique in a $k$-map graph is $\lfloor 3k/2 \rfloor$, the recognition of $k$-map graphs is still open for any fixed $k \ge 5$.