Polynomial kernelization for removing induced claws and diamonds

Abstract

A graph is called {claw, diamond}-free if it contains neither a claw (a K1,3) nor a diamond (a K4 with an edge removed) as an induced subgraph, or, equivalently, it is a line graph of a triangle-free graph. 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 also show that, even on instances with maximum degree 6, the problem is NP-complete and cannot be solved in time 2o(k) ・ |V (G)|O(1), assuming the Exponential Time Hypothesis.