Induced Disjoint Paths in Claw-Free Graphs

Abstract

Paths $P_1,\ldots,P_k$ in a graph $G=(V,E)$ are said to be mutually induced if for any $1\leq i<j \leq k$, $P_i$ and $P_j$ have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to test whether a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that $P_i$ connects $s_i$ and $t_i$ for $i=1,\ldots,k$. We show that this problem is fixed-parameter tractable for claw-free graphs when parameterized by $k$. Several related problems, such as the $k$-in-a-Path problem, are proved to be fixed-parameter tractable for claw-free graphs as well. We show that an improvement of these results in certain directions is unlikely, for example, by noting that the Induced Disjoint Paths problem cannot have a polynomial kernel for line graphs (a type of claw-free graphs), unless \sf NP $\subseteq$ co${\sf NP}/$poly. Moreover, the problem becomes \sf NP-complete, even when $k=2$, for the more general cla...