Induced Disjoint Paths in AT-Free Graphs

Abstract

Paths P1,. .. , P k in a graph G = (V, E) are mutually induced if any two distinct Pi and Pj 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 (si, ti) contains k mutually induced paths Pi such that each Pi connects si and ti. 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 ATfree 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 ATfree 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 |VH |, even on a subclass of AT-free graph, namely cobipartite graphs. We also show that the problems kin a Path and kin a Tree are polynomialtime 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.

Graph Editing to a Fixed Target

“…It is known that for all graphs H, the HInduced Topological Minor problem is polynomial-time solvable for ATfree graphs [12], chordal graphs [2] and planar graphs [19]. However, as noted in Section 2, we cannot always guarantee the existence of a nice topological minor sequence of sufficiently small length.…”

Parameterized Complexity of Induced H-Matching on Claw-Free Graphs

“…Unfortunately, extending to the case where h is also a parameter is unlikely to succeed, as Induced H-Matching becomes W[1]-hard even on line graphs and co-bipartite graphs [13,14].…”

“…The main result of this paper is that Induced H-Matching is fixed-parameter tractable on claw-free graphs when parameterized by k for fixed connected graphs H of constant size. It is important to note that requiring H to be of constant size is essential, since the problem is W[1]-hard with respect to k + |V (H)| even for line graphs and co-bipartite graphs [13,14]. In the special case that H is a fixed clique, we also show that the problem admits a polynomial kernel.…”

The (theta, wheel)-free graphs Part II: Structure theorem

“…It can be solved for line graphs as a simple corollary of the linkage algorithm mentioned above. For any fixed k, it can be solved in linear time for planar graphs [16] and circular-arc graphs [14], and in polynomial time for AT-free graphs [12] and claw-free graphs [9].…”