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

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Abstract

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 \(u,v\in S\), 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 \(\frac{8}{15}\) (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 \(\frac{3v(G)-8}{4}\) vertices, where v(G) denotes the number of vertices of G.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

We would like to give our thanks to anonymous reviewers for careful reading of this paper and many valuable suggestions.

Funding

This work was supported by NSFC (Grant No. 11771080).

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  1. School of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Wenying Xi & Wensong Lin

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  1. Wenying Xi
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  2. Wensong Lin
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Correspondence to Wensong Lin.

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Xi, W., Lin, W. The maximum 3-star packing problem in claw-free cubic graphs. J Comb Optim 47, 73 (2024). https://doi.org/10.1007/s10878-024-01115-z

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