Induced Disjoint Paths in Claw-Free Graphs

  • Petr A. Golovach
  • Daniël Paulusma
  • Erik Jan van Leeuwen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7501)


Paths P 1,…,P k in a graph G = (V,E) are said to be mutually induced if for any 1 ≤ i < j ≤ 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,…,k. This problem is known to be NP-complete already for k = 2, but for n-vertex claw-free graphs, Fiala et al.gave an n O(k)-time algorithm. We improve the latter result by showing that the 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 shown to be fixed-parameter tractable for claw-free graphs as well. We prove 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 NP ⊆ coNP/poly. Moreover, the problem becomes NP-complete, even when k = 2, for the more general class of K 1,4-free graphs. Finally, we show that the n O(k)-time algorithm of Fiala et al.for testing whether a claw-free graph contains some k-vertex graph H as a topological induced minor is essentially optimal by proving that this problem is W[1]-hard even if G and H are line graphs.


Line Graph Interval Graph Polynomial Kernel Disjoint Path Graph Class 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Petr A. Golovach
    • 1
  • Daniël Paulusma
    • 1
  • Erik Jan van Leeuwen
    • 2
  1. 1.School of Engineering and Computer ScienceDurham UniversityEngland
  2. 2.Dept. Computer, Control, Managm. Eng.Sapienza University of RomeItaly

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