On Polynomial Kernelization of $\mathcal {H}$-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:

• $\mathcal {H}$ contains $K_{1,s}$ and $K_t$;

• $\mathcal {H}$ contains $K_{1,s}$ and the input graphs are $K_t$-free;

• $\mathcal {H}$ contains $K_{t}$ and the input graphs are $K_{1,s}$-free;

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.