# Contracting edges to destroy a pattern: A complexity study

## TL;DR

This paper studies the computational complexity of contracting edges in a graph to eliminate a certain pattern, showing NP-completeness in most cases and analyzing fixed-parameter tractability for H-free contractions.

## Contribution

It characterizes the complexity of H-free Contraction problems, proving NP-completeness for all but trivial cases and analyzing fixed-parameter tractability for tree patterns.

## Key findings

- H-free Contraction is polynomial-time solvable only when H is a small complete graph.
- Most H-free Contraction problems are NP-complete.
- H-free Contraction is W[2]-hard for most trees, with a few exceptions.

## Abstract

Given a graph G and an integer k, the objective of the $\Pi$-Contraction problem is to check whether there exists at most k edges in G such that contracting them in G results in a graph satisfying the property $\Pi$. We investigate the problem where $\Pi$ is `H-free' (without any induced copies of H). It is trivial that H-free Contraction is polynomial-time solvable if H is a complete graph of at most two vertices. We prove that, in all other cases, the problem is NP-complete. We then investigate the fixed-parameter tractability of these problems. We prove that whenever H is a tree, except for seven trees, H-free Contraction is W[2]-hard. This result along with the known results leaves behind three unknown cases among trees.