On minimum t-claw deletion in split graphs

Abstract

For \(t\ge 3\), \(K_{1, t}\) is called t-claw. A graph \(G=(V, E)\) is t-claw free if it does not contain t-claw as a vertex-induced subgraph. In minimum t-claw deletion problem (Min-t-Claw-Del), given a graph \(G=(V, E)\), it is required to find a vertex set S of minimum size such that \(G[V\setminus S]\) is t-claw free. In a split graph, the vertex set is partitioned into two sets such that one forms a clique and the other forms an independent set. Every t-claw in a split graph has a center vertex in the clique partition. This observation motivates us to consider the minimum one-sided bipartite t-claw deletion problem (Min-t-OSBCD). Given a bipartite graph \(G=(A \cup B, E)\), in Min-t-OSBCD it is asked to find a vertex set S of minimum size such that \(G[(A \cup B) {\setminus } S]\) has no t-claw with the center vertex in A. A primal-dual algorithm approximates Min-t-OSBCD within a factor of t. We prove that it is \({\textsf{UGC}}\)-hard to approximate with a factor better than t. We also prove it is approximable within a factor of 2 for dense bipartite graphs. By using these results on Min-t-OSBCD, we prove that Min-t-Claw-Del is \({\textsf{UGC}}\)-hard to approximate within a factor better than t, for split graphs. We also consider their complementary maximization problems and prove that they are \({\textsf{APX}}\)-complete.