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].