Contracting to a Longest Path in H-Free Graphs

TL;DR

This paper classifies the computational complexity of finding long paths in H-free graphs, introducing a new contraction technique that reduces the problem to a matching problem, advancing understanding of graph pattern detection.

Contribution

It provides a complete complexity classification for Longest Path Contractibility in H-free graphs and introduces a novel contraction method reducing the problem to matching.

Findings

NP-hardness results for Longest Path Contractibility in H-free graphs

A new contraction technique reducing the problem to matching

Complete classification of complexity for both problems

Abstract

We prove two dichotomy results for detecting long paths as patterns in a given graph. The NP-hard problem Longest Induced Path is to determine the longest induced path in a graph. The NP-hard problem Longest Path Contractibility is to determine the longest path to which a graph can be contracted to. By combining known results with new results we completely classify the computational complexity of both problems for $H$-free graphs. Our main focus is on the second problem, for which we design a general contractibility technique that enables us to reduce the problem to a matching problem.