Minimum fill-in and treewidth of [formula omitted] and [formula omitted] graphs

Abstract

In this paper we investigate how graph problems that are NP-hard in general, but polynomial time solvable on split graphs, behave on input graphs that are close to being split. For this purpose we define split + k e and split + k v graphs to be the graphs that can be made split by removing at most k edges and at most k vertices, respectively. We show that problems like treewidth and minimum fill-in are fixed parameter tractable with parameter k on split + k e graphs. Along with positive results of fixed parameter tractability of several problems on split + k e and split + k v graphs, we also show a surprising hardness result. We prove that computing the minimum fill-in of split + k v graphs is NP-hard even for k = 1 . This implies that also minimum fill-in of chordal + k v graphs is NP-hard for every k . In contrast, we show that the treewidth of split + 1 v graphs can be computed in linear time. This gives probably the first graph class for which the treewidth and the minimum fill-in problems have different computational complexity.