SWAT 2012: Algorithm Theory – SWAT 2012 pp 153-164

# Induced Disjoint Paths in AT-Free Graphs

• Petr A. Golovach
• Daniël Paulusma
• Erik Jan van Leeuwen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)

## Abstract

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. 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 |V H |. 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.

## Keywords

Polynomial Time Planar Graph Interval Graph Disjoint Path Chordal Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 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 Computing SciencesDurham University, Science LaboratoriesDurhamUK
2. 2.Dept. Computer and System SciencesUniversity of Rome “La Sapienza”RomaItaly