Incompressibility of H-Free Edge Modification

Abstract

Given a fixed graph H, the H -Free Edge Deletion (resp., Completion, Editing) problems ask whether it is possible to delete from (resp., add to, delete from or add to) the input graph at most k edges so that the resulting graph is H-free, i.e., contains no induced subgraph isomorphic to H. These H-free edge modification problems are well known to be FPT for every fixed H. In this paper, we study the nonexistence of polynomial kernels for them in terms of the structure of H, and completely characterize their nonexistence for H being paths, cycles or 3-connected graphs. As a very effective tool, we have introduced a constrained satisfiability problem Propagational Satisfiability to cope with the propagation of edge additions/deletions, and we expect the problem to be useful in studying the nonexistence of polynomial kernels.