Induced Disjoint Paths and Connected Subgraphs for $H$-Free Graphs

TL;DR

This paper studies the computational complexity of finding induced disjoint paths and connected subgraphs in H-free graphs, providing dichotomies and classifications for these problems.

Contribution

It introduces a generalization of the classical problem and offers a comprehensive complexity classification for H-free graphs, including fixed parameter cases.

Findings

Almost-complete dichotomies for H-free graphs

Complete classification for fixed number of terminal sets

Complexity results for both problems in various graph classes

Abstract

Paths $P_1,\ldots, P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that each $P_i$ starts from $s_i$ and ends at $t_i$. This is a classical graph problem that is NP-complete even for $k=2$. We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give almost-complete dichotomies of the computational complexity of both problems for H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. Finally, we give a complete classification of the complexity of the second problem if the number k of terminal sets is fixed, that is, not part of the input.