Incompressibility of $$H$$ H -Free Edge Modification Problems

Abstract

Given a fixed graph $$H$$H, the $$H$$H-Free Edge Deletion (resp., Completion, Editing) problem asks whether it is possible to delete from (resp., add to, delete from or add to) the input graph at most $$k$$k edges so that the resulting graph is $$H$$H-free, i.e., contains no induced subgraph isomorphic to $$H$$H. These $$H$$H-free edge modification problems are well known to be fixed-parameter tractable for every fixed $$H$$H. In this paper we study the incompressibility, i.e., nonexistence of polynomial kernels, for these $$H$$H-free edge modification problems in terms of the structure of $$H$$H, and completely characterize their nonexistence for $$H$$H being paths, cycles or 3-connected graphs. We also give a sufficient condition for the nonexistence of polynomial kernels for $${\mathcal {F}}$$F-Free Edge Deletion problems, where $${\mathcal {F}}$$F is a finite set of forbidden induced subgraphs. As an effective tool, we have introduced an incompressible constraint satisfiability problem Propagational-$$f$$f Satisfiability to express common propagational behaviors of events, and we expect the problem to be useful in studying the nonexistence of polynomial kernels in general.