Induced Disjoint Paths in AT-free Graphs
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Paths P_1,…,P_k in a graph G=(V,E) are mutually induced if any two distinct P_i and P_j have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (s_i,t_i) contains k mutually induced paths P_i such that each P_i connects s_i and t_i. This is a classical graph problem that is NP-complete even for k=2. We study it for AT-free graphs. Unlike its subclasses of permutation graphs and cocomparability graphs, the class of AT-free graphs has no geometric intersection model. However, by a new, structural analysis of the behaviour of Induced Disjoint Paths for AT-free graphs, we prove that it can be solved in polynomial time for AT-free graphs even when k is part of the input. This is in contrast to the situation for other well-known graph classes, such as planar graphs, claw-free graphs, or more recently, (theta,wheel)-free graphs, for which such a result only holds if k is fixed. As a consequence of our main result, the problem of deciding if a given AT-free graph contains a fixed graph H as an induced topological minor admits a polynomial-time algorithm. In addition, we show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard with parameter |V_H|, even on a subclass of AT-free graph, namely cobipartite graphs. We also show that the problems k-in-a-Path and k-in-a-Tree are polynomial-time solvable on AT-free graphs even if k is part of the input. These problems are to test if a graph has an induced path or induced tree, respectively, spanning k given vertices.
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