The maximum 3-star packing problem in claw-free cubic graphs

A 3-star is a complete bipartite graph K 1 , 3 . A 3-star packing of a graph G is a collection of vertex-disjoint subgraphs of G in which each subgraph is a 3-star. The maximum 3-star packing problem is to find a 3-star packing of a given graph with the maximum number of 3-stars. A 2 -independent set of a graph G is a subset S of V ( G ) such that for each pair of vertices u , v ∈ S , paths between u and v are all of length at least 3. In cubic graphs, the maximum 3-star packing problem is equivalent to the maximum 2-independent set problem. The maximum 2-independent set problem was proved to be NP-hard on cubic graphs (Kong and Zhao in Congressus Numerantium 143:65–80, 2000), and the best approximation algorithm of maximum 2-independent set problem for cubic graphs has approximation ratio 8 15 (Miyano et al. in WALCOM 2017, Proceedings, pp 228–240). In this paper, we first prove that the maximum 3-star packing problem is NP-hard in claw-free cubic graphs and then design a linear-time algorithm which can find a 3-star packing of a connected claw-free cubic graph G covering at least 3 v ( G ) - 8 4 vertices, where v ( G ) denotes the number of vertices of G .