# Decomposition of Map Graphs with Applications

## TL;DR

This paper introduces a new decomposition lemma for map graphs that enables the design of subexponential-time parameterized algorithms for various problems, extending bidimensionality techniques to this class.

## Contribution

The authors prove a novel decomposition lemma for map graphs and demonstrate its application in developing subexponential algorithms for several problems.

## Key findings

- Subexponential algorithms for Connected Planar F-Deletion on map graphs.
- First subexponential algorithms for Longest Cycle/Path on map graphs.
- Improved algorithms for Feedback Vertex Set and Cycle Packing on map graphs.

## Abstract

Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and Thomas that states that every planar graph either has a $\sqrt{k}\times \sqrt{k}$-grid as a minor, or its treewidth is $O(\sqrt{k})$. However, bidimensionality theory cannot be extended directly to several well-known classes of geometric graphs. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Inspired by this, we prove a new decomposition lemma for map graphs. Informally, our lemma states the following. For any map graph $G$, there exists a collection $(U_1,\ldots,U_t)$ of cliques of $G$ with the following property: $G$ either contains a $\sqrt{k}\times \sqrt{k}$-grid as a minor, or it admits a tree decomposition where every bag is the union of $O(\sqrt{k})$ of the cliques in the above collection. The new lemma appears to be a handy tool in the design of subexponential parameterized algorithms on map graphs. We demonstrate its usability by designing algorithms on map graphs with running time $2^{O({\sqrt{k}\log{k}})} \cdot n^{O(1)}$ for the Connected Planar $\cal F$-Deletion problem (that encompasses problems such as Feedback Vertex Set and Vertex Cover). Obtaining subexponential algorithms for Longest Cycle/Path and Cycle Packing is more challenging. We have to construct tree decompositions with more powerful properties and to prove sublinear bounds on the number of ways an optimum solution could "cross" bags in these decompositions. For Longest Cycle/Path, these are the first subexponential-time parameterized algorithms on map graphs. For Feedback Vertex Set and Cycle Packing, we improve upon known $2^{O({k^{0.75}\log{k}})} \cdot n^{O(1)}$-time algorithms on map graphs.