(Sub)linear Kernels for Edge Modification Problems Toward Structured Graph Classes

In a (parameterized) graph edge modification problem, we are given G, an integer k and (usually well-structured) class $$\mathcal {G}$$ of graphs, asked whether it is possible to transform G into $$G' \in \mathcal by adding and/or removing at most edges. Parameterized problems received considerable attention in the last decades. this paper, focus on finding small kernels for problems. One studied Cluster Editing which goal partition vertex set disjoint union cliques. Even if 2k-vertex kernel exists Editing, does not reduce size instance cases. Therefore, explore question linear theoretical limit problems, particular when target very structured (such as cliques instance). We prove, far know, first sublinear problem. Namely, show that Clique + Independent Set Deletion, restriction admits $$O(k/\log k)$$ . also obtain several other Deletion improving previous 4k-vertex kernel. prove (Pseudo-)Split Completion (and equivalent Deletion) kernel, existing quadratic Trivially Perfect (improving cubic kernel), finally its triangle-free version (Starforest optimal under Exponential Time Hypothesis.