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

Abstract

For a set of graphs $\mathcal {H}$, the $\mathcal {H}$ -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 [3], it is shown that the problem is fixed-parameter tractable if $\mathcal {H}$ is of finite cardinality. However, it is proved in [4] 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 $\mathcal {H}$ -free Edge Deletion problems which remain NP-complete even for the bounded degree input graphs, for example Triangle-free Edge Deletion [2] and Custer Edge Deletion( $P_3$ -free Edge Deletion) [15]. 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 $K_{1,s}$ -free Edge Deletion for general graphs [5]. Our result provides first polynomial kernels for Claw-free Edge Deletion and Line Edge Deletion for $K_t$-free input graphs which are NP-complete even for $K_4$-free graphs [23] and were raised as open problems in [4, 19].

R.B. Sandeep—supported by TCS Research Scholarship.