Linear-Time Algorithms for Eliminating Claws in Graphs

Since many \({\mathsf {NP}}\)-complete graph problems have been shown polynomial-time solvable when restricted to claw-free graphs, we study the problem of determining the distance of a given graph to a claw-free graph, considering vertex elimination as measure. Claw-free Vertex Deletion (CFVD) consists of determining the minimum number of vertices to be removed from a graph such that the resulting graph is claw-free. Although CFVD is \({\mathsf {NP}}\)-complete in general and recognizing claw-free graphs is still a challenge, where the current best deterministic algorithm consists of performing |V(G)| executions of the best algorithm for matrix multiplication, we present linear-time algorithms for CFVD on weighted block graphs and weighted graphs with bounded treewidth. Furthermore, we show that this problem can be solved in linear time by a simpler algorithm on forests, and we determine the exact values for full k-ary trees. On the other hand, we show that Claw-free Vertex Deletion is \({\mathsf {NP}}\)-complete even when the input graph is a split graph. We also show that the problem is hard to approximate within any constant factor better than 2, assuming the Unique Games Conjecture.