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

Abstract

For a set $$\mathcal {H}$$ of graphs, the $$\mathcal {H}$$ -free Edge Deletion problem is to decide whether there exist at most k edges in the input graph, for some $$k\in \mathbb {N}$$ , whose deletion results in a graph without an induced copy of any of the graphs in $$\mathcal {H}$$ . The problem is known to be fixed-parameter tractable if $$\mathcal {H}$$ is of finite cardinality. In this paper, we present a polynomial kernel for this problem for any fixed finite set $$\mathcal {H}$$ of connected graphs for the case where the input graphs are of bounded degree. We use a single kernelization rule which deletes vertices ‘far away’ from the induced copies of every $$H\in \mathcal {H}$$ in the input graph. With a slightly modified kernelization rule, we obtain polynomial kernels for $$\mathcal {H}$$ -free Edge Deletion under the following three settings: where $$s>1$$ and $$t>2$$ are any fixed integers. Our result provides the first polynomial kernels for Claw-free Edge Deletion and Line Edge Deletion for $$K_t$$ -free input graphs which are known to be NP-complete even for $$K_4$$ -free graphs.