(Sub)linear kernels for edge modification problems towards structured graph classes

In a (parameterized) graph edge modification problem, we are given a graph $G$, an integer $k$ and a (usually well-structured) class of graphs $\mathcal{G}$, and ask whether it is possible to transform $G$ into a graph $G' \in \mathcal{G}$ by adding and/or removing at most $k$ edges. Parameterized graph edge modification problems received considerable attention in the last decades. In this paper we focus on finding small kernels for edge modification problems. One of the most studied problems is the \textsc{Cluster Editing} problem, in which the goal is to partition the vertex set into a disjoint union of cliques. Even if a $2k$ kernel exists for \textsc{Cluster Editing}, this kernel does not reduce the size of the instance in most cases. Therefore, we explore the question of whether linear kernels are a theoretical limit in edge modification problems, in particular when the target graphs are very structured (such as a partition into cliques for instance). We prove, as far as we know, the first sublinear kernel for an edge modification problem. Namely, we show that \textsc{Clique + Independent Set Deletion}, which is a restriction of \textsc{Cluster Deletion}, admits a kernel of size $O(k/\log k)$. We also obtain small kernels for several other edge modification problems. We prove that \textsc{Split Addition} (and the equivalent \textsc{Split Deletion}) admits a linear kernel, improving the existing quadratic kernel of Ghosh et al. \cite{ghosh2015faster}. We also prove that \textsc{Trivially Perfect Addition} admits a quadratic kernel (improving the cubic kernel of Guo \cite{guo2007problem}), and finally prove that its triangle-free version (\textsc{Starforest Deletion}) admits a linear kernel, which is optimal under ETH.