Incompressibility of $$$$-Free Edge Modification Problems

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.