On Polynomial Kernelization of $\mathcal{H}$-free Edge Deletion

For a set of graphs $\mathcal{H}$, the \textsc{$\mathcal{H}$-free Edge Deletion} problem asks to find whether there exist at most $k$ edges in the input graph whose deletion results in a graph without any induced copy of $H\in\mathcal{H}$. In \cite{cai1996fixed}, it is shown that the problem is fixed-parameter tractable if $\mathcal{H}$ is of finite cardinality. However, it is proved in \cite{cai2013incompressibility} that if $\mathcal{H}$ is a singleton set containing $H$, for a large class of $H$, there exists no polynomial kernel unless $coNP\subseteq NP/poly$. In this paper, we present a polynomial kernel for this problem for any fixed finite set $\mathcal{H}$ of connected graphs and when the input graphs are of bounded degree. We note that there are \textsc{$\mathcal{H}$-free Edge Deletion} problems which remain NP-complete even for the bounded degree input graphs, for example \textsc{Triangle-free Edge Deletion}\cite{brugmann2009generating} and \textsc{Custer Edge Deletion($P_3$-free Edge Deletion)}\cite{komusiewicz2011alternative}. When $\mathcal{H}$ contains $K_{1,s}$, we obtain a stronger result - a polynomial kernel for $K_t$-free input graphs (for any fixed $t> 2$). We note that for $s>9$, there is an incompressibility result for \textsc{$K_{1,s}$-free Edge Deletion} for general graphs \cite{cai2012polynomial}. Our result provides first polynomial kernels for \textsc{Claw-free Edge Deletion} and \textsc{Line Edge Deletion} for $K_t$-free input graphs which are NP-complete even for $K_4$-free graphs\cite{yannakakis1981edge} and were raised as open problems in \cite{cai2013incompressibility,open2013worker}.