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

Author

Listed:
  • Wenying Xi

    (Southeast University)

  • Wensong Lin

    (Southeast University)

Abstract

A 3-star is a complete bipartite graph $$K_{1,3}$$ 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 $$u,v\in S$$ 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 $$\frac{8}{15}$$ 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 $$\frac{3v(G)-8}{4}$$ 3 v ( G ) - 8 4 vertices, where v(G) denotes the number of vertices of G.

Suggested Citation

  • Wenying Xi & Wensong Lin, 2024. "The maximum 3-star packing problem in claw-free cubic graphs," Journal of Combinatorial Optimization, Springer, vol. 47(5), pages 1-16, July.
    • Handle: RePEc:spr:jcomop:v:47:y:2024:i:5:d:10.1007_s10878-024-01115-z
    DOI: 10.1007/s10878-024-01115-z
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