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On the d-Claw Vertex Deletion Problem

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Abstract

Let d-claw (or d-star) stand for \(K_{1,d}\), the complete bipartite graph with 1 and \(d\ge 1\) vertices on each part. The d-claw vertex deletion problem, \(d\)-claw-vd, asks for a given graph G and an integer k if one can delete at most k vertices from G such that the resulting graph has no d-claw as an induced subgraph. Thus, \(1\)-claw-vd and \(2\)-claw-vd are just the famous vertex cover problem and the cluster vertex deletion problem, respectively. In this paper, we strengthen a hardness result recently proved in Jena and Subramani (in: Du, Du, Wu, and Zhu (eds) Theory and applications of models of computation - 17th annual conference, TAMC 2022, Tianjin, China, September 16–18, 2022, Proceedings, 2022), by showing that cluster vertex deletion remains \(\textsf{NP}\)-complete even when restricted to planar bipartite graphs of maximum degree 3 and arbitrary large girth. Moreover, for every \(d\ge 3\), we show that \(d\)-claw-vd is \(\textsf{NP}\)-complete even when restricted to planar bipartite graphs of maximum degree d. These hardness results are optimal with respect to degree constraint. By extending the hardness result in Bonomo-Braberman et al (in: Computing and combinatorics - 26th international conference, COCOON 2020, Proceedings, Lecture Notes in Computer Science, vol 12273, Springer, 2020, pp 14–26, 2020), we show that, for every \(d\ge 3\), \(d\)-claw-vd is \(\textsf{NP}\)-complete even when restricted to split graphs without \((d+1)\)-claws, and split graphs of diameter 2. On the positive side, we prove that \(d\)-claw-vd is polynomially solvable on what we call d-block graphs, a class properly contains all block graphs. This result extends the polynomial-time algorithm in Cao et al (Theor Comput Sci, 2018) for \(2\)-claw-vd on block graphs to \(d\)-claw-vd for all \(d\ge 2\) and improves the polynomial-time algorithm proposed by Bonomo-Brabeman et al. for (unweighted) \(3\)-claw-vd on block graphs to 3-block graphs.

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Notes

  1. The \(O^*\) notation hides polynomial factors.

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Acknowledgements

We thank Ling-Ju Hung (NTUB, Taoyuan, Taiwan) for pointing the paper [14] to us.

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Authors and Affiliations

  1. Department of Computer Science and Information Engineering, National Cheng Kung University, No. 1, University Road, Tainan, 70101, Taiwan

    Sun-Yuan Hsieh

  2. Berlin, Germany

    Hoang-Oanh Le

  3. Institut für Informatik, Universität Rostock, Rostock, Germany

    Van Bang Le

  4. Department of Creative Technologies and Product Design, National Taipei University of Business, Taoyuan, 32462, Taiwan

    Sheng-Lung Peng

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  1. Sun-Yuan Hsieh
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A preliminary version [13] of this article has appeared in Proceedings of the 26th International Computing and Combinatorics Conference (COCOON 2021), Lecture Notes in Computer Science 13025, pp. 591–603.

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Hsieh, SY., Le, HO., Le, V.B. et al. On the d-Claw Vertex Deletion Problem. Algorithmica 86, 505–525 (2024). https://doi.org/10.1007/s00453-023-01144-w

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