Polynomial Kernelization for Removing Induced Claws and Diamonds

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 are characterized as line graphs of triangle-free graphs, or as linear dominoes (graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique). 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 result by proving that, even on instances with maximum degree 6, the problem is NP-complete and cannot be solved in time 2 o(k) · |V (G)| O(1) unless the Exponential Time Hypothesis fails.