Contracting planar graphs to contractions of triangulations

TL;DR

This paper identifies a class of graphs for which contracting a planar graph to any graph in this class can be decided efficiently, with the complexity depending only on the size of the target graph.

Contribution

The paper characterizes a class of graphs closed under contractions where contraction problems are fixed-parameter tractable for planar graphs.

Findings

The class $\\cal C$ is the closure of planar triangulated graphs under contractions.

A graph $H$ belongs to $\cal C$ iff it is contained as a contraction in any graph with sufficiently large tree-width.

Provides a characterization of $\cal C$ via minimal forbidden contractions.

Abstract

For every graph $H$, there exists a polynomial-time algorithm deciding if a planar input graph $G$ can be contracted to~$H$. However, the degree of the polynomial depends on the size of $H$. In this paper, we identify a class of graphs $\cal C$ such that for every $H \in \cal C$, there exists an algorithm deciding in time $f(|V(H)|) \cdot |V(G)|^{\bigO{1}}$ whether a planar graph $G$ can be contracted to~$H$. (The function $f(\cdot)$ does not depend on $G$.) The class $\cal C$ is the closure of planar triangulated graphs under taking of contractions. In fact, we prove that a graph $H \in \cal C$ if and only if there exists a constant $c_H$ such that if the tree-width of a graph is at least $c_H$, it contains $H$ as a contraction. We also provide a characterization of $\cal C$ in terms of minimal forbidden contractions.