Finding Disjoint Paths in Split Graphs

Abstract

The well-known DISJOINT PATHS problem takes as input a graph G and a set of k pairs of terminals in G, and the task is to decide whether there exists a collection of k pairwise vertex-disjoint paths in G such that the vertices in each terminal pair are connected to each other by one of the paths. This problem is known to be NP-complete, even when restricted to planar graphs or interval graphs. Moreover, although the problem is fixed-parameter tractable when parameterized by k due to a celebrated result by Robertson and Seymour, it is known not to admit a polynomial kernel unless NP ⊆ coNP/poly. We prove that DISJOINT PATHS remains NP-complete on split graphs, and show that the problem admits a kernel with O(k 2 ) vertices when restricted to this graph class. We furthermore prove that, on split graphs, the edge-disjoint variant of the problem is also NP-complete and admits a kernel with O(k 3 ) vertices. To the best of our knowledge, our kernelization results are the first non-trivial kernelization results for these problems on graph classes.

Disjoint paths and connected subgraphs for H-free graphs

“…As Disjoint Paths is NP-complete, it is natural to consider special graph classes. The Disjoint Paths problem is known to be NP-complete even for graph of clique-width at most 6 [8], split graphs [9], interval graphs [15] and line graphs. The latter result can be obtained by a straightforward reduction (see, for example, [8,9]) from its edge variant, Edge Disjoint Paths, proven to be NP-complete by Even, Itai and Shamir [5].…”

“…The Disjoint Paths problem is known to be NP-complete even for graph of clique-width at most 6 [8], split graphs [9], interval graphs [15] and line graphs. The latter result can be obtained by a straightforward reduction (see, for example, [8,9]) from its edge variant, Edge Disjoint Paths, proven to be NP-complete by Even, Itai and Shamir [5]. On the positive side, Disjoint Paths is polynomial-time solvable for cographs, or equivalently, P 4 -free graphs [8].…”

“…The first lemma of a series of four is obtained by a straightforward reduction from the Edge Disjoint Paths problem (see, e.g. [8,9]), which was proven to be NP-complete by Even, Itai and Shamir [5]. The second lemma follows from the observation that an edge subdivision of the graph G in an instance of Disjoint Paths results in an equivalent instance of Disjoint Paths; we apply this operation a sufficiently large number of times to obtain a graph of large girth.…”

Induced disjoint paths in circular-arc graphs in linear time

“…The hardness of both Disjoint Paths and Induced Disjoint Paths on general graphs inspired research on their complexity on structured graph classes. On the negative side, Disjoint Paths remains NP-complete on line graphs [20] and split graphs [14], and Induced Disjoint Paths remains NP-complete on claw-free graphs [6] (in fact, even on line graphs). Both problems remain NPcomplete on planar graphs [19,8].…”

“…Both problems remain NPcomplete on planar graphs [19,8]. In these cases, however, fixed-parameter algorithms are known [9,14,17,24,25]. On the positive side, polynomial-time algorithms for Disjoint Paths exist on graphs of bounded treewidth [23] and graphs of clique-width at most 2 [12], and for Induced Disjoint Paths on AT-free graphs [8] and chordal graphs [1].…”