Parameterized Complexity of Vertex Deletion into Perfect Graph Classes

Vertex deletion problems are at the heart of parameterized complexity. For a graph class F , the F-Deletion problem takes as input a graph G and an integer k. The question is whether it is possible to delete at most k vertices from G such that the resulting graph belongs to F . Whether Perfect Deletion is fixed-parameter tractable, and whether Chordal Deletion admits a polynomial kernel, when parameterized by k, have been stated as open questions in previous work. We show that Perfect Deletion and Weakly Chordal Deletion are W [2]-hard when parameterized by k. In search of positive results, we study a restricted variant of the F-Deletion problem. In this restricted variant, the deleted vertices must be taken from a specified set X, and we parameterize by |X|. We show that for Perfect Deletion and Weakly Chordal Deletion, although this restriction immediately ensures fixed-parameter tractability, it is not enough to yield polynomial kernels, unless NP ⊆ coNP/poly. On the positive side, for Chordal Deletion, the restriction enables us to obtain a kernel with O(|X|) vertices. keywords: parameterized complexity; perfect graphs; vertex deletion problems; kernelization