Further parameterized algorithms for the [formula omitted]-free edge deletion problem

Abstract

Given a graph G=(V,E) and a set F of forbidden subgraphs, we study the F-Free Edge Deletion problem, where the goal is to remove a minimum number of edges such that the resulting graph does not contain any F∈F as a (not necessarily induced) subgraph. Enright and Meeks (Algorithmica, 2018) gave an algorithm to solve F-Free Edge Deletion whose running time on an n-vertex graph G of treewidth tw(G) is bounded by 2O(|F|tw(G)r)n, if every graph in F has at most r vertices. We complement this result by showing that F-Free Edge Deletion is W[1]-hard when parameterized by tw(G)+|F|. We also show that F-Free Edge Deletion is W[2]-hard when parameterized by the combined parameters solution size, the feedback vertex set number and pathwidth of the input graph. A special case of particular interest is the situation in which F is the set Th+1 of all trees on h+1 vertices, so that we delete edges in order to obtain a graph in which every component contains at most h vertices. This is desirable from the point of view of restricting the spread of a disease in transmission networks [5]. We prove that Th+1-Free Edge Deletion is fixed-parameter tractable (FPT) when parameterized by the vertex cover number of the input graph. We also prove that it admits a kernel with 2kh vertices and 2kh2+k edges, when parameterized by k+h.