Incompressibility of H-free edge modification problems: Towards a dichotomy
Dániel Marx, R. B. Sandeep · Journal of Computer and System Sciences · 2021
Given a graph G and an integer k, the H-free Edge Editing problem is to find whether there exist at most k pairs of vertices in G such that changing the adjacency of the pairs in G results in a graph without any induced copy of H. Nontrivial polynomial kernels are known to exist for some graphs H with at most 4 vertices, but starting from 5 vertices, polynomial kernels are known only if H is either complete or empty. This suggests the conjecture that there is no other H with at least 5 vertices where H-free Edge Editing admits a polynomial kernel. Towards this goal, we obtain a set H of nine 5-vertex graphs such that if for every H∈H, H-free Edge Editing is incompressible and the complexity assumption NP⊈coNP/poly holds, then H-free Edge Editing is incompressible for every graph H with at least five vertices that is neither complete nor empty. We obtain similar results also for H-free Edge Deletion/Completion.