The Parameterized Landscape of Labeled Graph Contractions

TL;DR

This paper investigates the computational complexity of finding maximum common contractions between labeled graphs, analyzing various parameters and revealing surprising similarities and differences with related problems.

Contribution

It provides a detailed parameterized complexity analysis of the maximum common contraction problem and compares it with the labeled contractibility problem.

Findings

Maximum common contraction problem is W[1]-hard when parameterized by degeneracy and number of contractions.

Labeled contractibility problem is fixed-parameter tractable (FPT) under similar parameters.

Little difference in complexity status between the two problems, except in a specific parameter setting.

Abstract

In this work, we study the problem of computing a maximum common contraction of two vertex-labeled graphs, i.e. how to make them identical by contracting as little edges as possible in the two graphs. We study the problem from a parameterized complexity point of view, using parameters such as the maximum degree, the degeneracy, the clique-width or treewidth of the input graphs as well as the number of allowed contractions. We put this complexity in perspective with that of the labeled contractibility problem, i.e determining whether a labeled graph is a contraction of another. Surprisingly, our results indicate very little difference between these problems in terms of parameterized complexity status. We only prove their status to differ when parameterizing by both the degeneracy and the number of allowed contractions, showing W[1]-hardness of the maximum common contraction problem in this case, whereas the contractibility problem is FPT.