Polynomial kernelization for removing induced claws and diamonds

TL;DR

This paper proves that the (claw,diamond)-free Edge Deletion problem has a polynomial kernel, but remains NP-complete and unlikely to be fixed-parameter tractable even for graphs with maximum degree 6.

Contribution

It establishes polynomial kernelization for the problem and shows its NP-completeness and complexity bounds on bounded degree graphs.

Findings

The problem admits a polynomial kernel.

NP-complete even on graphs with maximum degree 6.

No subexponential FPT algorithm unless ETH fails.

Abstract

A graph is called (claw,diamond)-free if it contains neither a claw (a $K_{1,3}$) nor a diamond (a $K_4$ with an edge removed) as an induced subgraph. Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique. In this paper we consider the parameterized complexity of the (claw,diamond)-free Edge Deletion problem, where given a graph $G$ and a parameter $k$, the question is whether one can remove at most $k$ edges from $G$ to obtain a (claw,diamond)-free graph. Our main result is that this problem admits a polynomial kernel. We complement this finding by proving that, even on instances with maximum degree $6$, the problem is NP-complete and cannot be solved in time $2^{o(k)}\cdot |V(G)|^{O(1)}$ unless the Exponential Time Hypothesis fail