Induced disjoint paths in AT-Free graphs

  • Authors:

  • Petr A. Golovach;Daniël Paulusma;Erik Jan van Leeuwen

  • Affiliations:

  • School of Engineering and Computing Sciences, Durham University, Science Laboratories, Durham, UK;School of Engineering and Computing Sciences, Durham University, Science Laboratories, Durham, UK;Dept. Computer and System Sciences, University of Rome "La Sapienza", Roma, Italy

  • Venue:
  • SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
  • Year:
  • 2012

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Abstract

Paths P1,…,Pk in a graph G=(V,E) are said to be mutually induced if for any 1≤ij≤k, Pi and Pj 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 (si,ti) contains k mutually induced paths Pi such that Pi connects si and ti for i=1,…,k. This problem is known to be NP-complete already for k=2. We prove that it can be solved in polynomial time for AT-free graphs even when k is part of the input. As a consequence, the problem of testing whether a given AT-free graph contains some graph H as an induced topological minor admits a polynomial-time algorithm, as long as H is fixed; we show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard, even on a subclass of AT-free graphs, namely cobipartite graphs, when parameterized by |VH|. We also show that the problems k-in-a-Path and k-in-a-Tree can be solved in polynomial time, even when k is part of the input. These problems are to test whether a graph contains an induced path or induced tree, respectively, spanning k given vertices.