Incompressibility of $H$-Free Edge Modification Problems

Abstract

Given a fixed graph $H$, the $H$-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 $k$ 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 fixed-parameter tractable for every fixed $H$. In this paper we study the incompressibility, i.e., nonexistence of polynomial kernels, for these $H$-free edge modification problems in terms of the structure of $H$, and completely characterize their nonexistence for $H$ being paths, cycles or 3-connected graphs. We also give a sufficient condition for the nonexistence of polynomial kernels for ${\mathcal {F}}$-Free Edge Deletion problems, where ${\mathcal {F}}$ is a finite set of forbidden induced subgraphs. As an effective tool, we have introduced an incompressible constraint satisfiability problem Propagational-$f$ 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.