Disjoint Paths and Connected Subgraphs for H-Free Graphs

TL;DR

This paper classifies the computational complexity of disjoint paths and connected subgraphs problems in H-free graphs, extending known results and identifying exceptions, with implications for graph algorithms.

Contribution

It provides a complete complexity classification for k-Disjoint Connected Subgraphs and nearly-complete for Disjoint Connected Subgraphs in H-free graphs, expanding understanding of these problems.

Findings

Classified complexity of k-Disjoint Connected Subgraphs in H-free graphs.

Provided almost-complete classification for Disjoint Connected Subgraphs.

Identified exceptions to known polynomial-time solvability in specific graph classes.

Abstract

The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct pairs. We determine, with an exception of two cases, the complexity of the Disjoint Paths problem for $H$-free graphs. If $k$ is fixed, we obtain the $k$-Disjoint Paths problem, which is known to be polynomial-time solvable on the class of all graphs for every $k \geq 1$. The latter does no longer hold if we need to connect vertices from terminal sets instead of terminal pairs. We completely classify the complexity of $k$-Disjoint Connected Subgraphs for $H$-free graphs, and give the same almost-complete classification for Disjoint Connected Subgraphs for $H$-free graphs as for Disjoint Paths.