Incompressibility of H-free edge modification problems: Towards a dichotomy

• Published in: Journal of computer and system sciences Vol. 125; pp. 25 - 58
• Main Authors: Marx, Dániel, Sandeep, R.B.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-05-2022
• Subjects: Edge modification problems; H-free graphs; Incompressibility; H-free graphs; Incompressibility; Edge modification problems
• ISSN: 0022-0000; 1090-2724
• DOI: 10.1016/j.jcss.2021.11.001

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.