Few Induced Disjoint Paths for $H$-Free Graphs

TL;DR

This paper investigates the computational complexity of the $k$-Induced Disjoint Paths problem in $H$-free graphs, revealing new NP-completeness results and contrasting them with known dichotomies for the non-induced variant.

Contribution

It establishes new complexity results for $k$-Induced Disjoint Paths in $H$-free graphs, highlighting differences from the non-induced case and providing a complexity dichotomy for the problem.

Findings

NP-completeness for certain $H$-free graph classes

Polynomial-time solvability in some restricted cases

Comparison with complexity dichotomy for the non-induced variant

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. For a fixed integer $k$, the $k$-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 $t_i$. Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer $k$, a classical result from the literature states that even $2$-Induced Disjoint Paths is NP-complete. We prove new complexity results for $k$-Induced Disjoint Paths if the input is restricted to $H$-free graphs, that is, graphs without a fixed graph $H$ as an induced subgraph. We compare our results with a complexity dichotomy for Induced Disjoint Paths, the variant where $k$ is part of the input.