Algorithmica

, Volume 62, Issue 1–2, pp 499–519

# The k-in-a-Path Problem for Claw-free Graphs

• Jiří Fiala
• Marcin Kamiński
• Bernard Lidický
• Daniël Paulusma
Article

## Abstract

The k-in-a-Path problem is to test whether a graph contains an induced path spanning k given vertices. This problem is NP-complete in general graphs, already when k=3. We show how to solve it in polynomial time on claw-free graphs, when k is an arbitrary fixed integer not part of the input. As a consequence, also the k-Induced Disjoint Paths and the k-in-a-Cycle problem are solvable in polynomial time on claw-free graphs for any fixed k. The first problem has as input a graph G and k pairs of specified vertices (s i ,t i ) for i=1,…,k and is to test whether G contain k mutually induced paths P i such that P i connects s i and t i for i=1,…,k. The second problem is to test whether a graph contains an induced cycle spanning k given vertices. When k is part of the input, we show that all three problems are NP-complete, even for the class of line graphs, which form a subclass of the class of claw-free graphs.

## Keywords

Induced path Claw-free graph Polynomial time algorithm

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## Authors and Affiliations

• Jiří Fiala
• 1
• Marcin Kamiński
• 2
• Bernard Lidický
• 1
• Daniël Paulusma
• 3
Email author
1. 1.Faculty of Mathematics and Physics, DIMATIA and Institute for Theoretical Computer Science (ITI)Charles UniversityPragueCzech Republic
2. 2.Computer Science DepartmentUniversité Libre de BruxellesBrusselsBelgium
3. 3.Department of Computer Science, Science LaboratoriesUniversity of DurhamDurham DH1England