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s-Club Cluster Vertex Deletion on Interval and Well-Partitioned Chordal Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13453))

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Abstract

In this paper, we study the computational complexity of s -Club Cluster Vertex Deletion. Given a graph, s -Club Cluster Vertex Deletion (s -CVD) aims to delete the minimum number of vertices from the graph so that each connected component of the resulting graph has a diameter at most s. When \(s=1\), the corresponding problem is popularly known as Cluster Vertex Deletion (CVD). We provide a faster algorithm for s -CVD on interval graphs. For each \(s\ge 1\), we give an \(O(n(n+m))\)-time algorithm for s -CVD on interval graphs with n vertices and m edges. In the case of \(s=1\), our algorithm is a slight improvement over the \(O(n^3)\)-time algorithm of Cao et al. (Theor. Comput. Sci., 2018) and for \(s \ge 2\), it significantly improves the state-of-the-art running time \(\left( O\left( n^4\right) \right) \).

We also give a polynomial-time algorithm to solve CVD on well-partitioned chordal graphs, a graph class introduced by Ahn et al. (WG 2020) as a tool for narrowing down complexity gaps for problems that are hard on chordal graphs, and easy on split graphs. Our algorithm relies on a characterisation of the optimal solution and on solving polynomially many instances of the Weighted Bipartite Vertex Cover. This generalises a result of Cao et al. (Theor. Comput. Sci., 2018) on split graphs. We also show that for any even integer \(s\ge 2\), s -CVD is NP-hard on well-partitioned chordal graphs.

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Author information

Authors and Affiliations

  1. Laboratoire de l’Informatique du Parallélisme, ENS de Lyon, Lyon, France

    Dibyayan Chakraborty

  2. Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India

    L. Sunil Chandran & Raji R. Pillai

  3. Department of Mathematics, BITS-Pilani, Hyderabad, India

    Sajith Padinhatteeri

Authors
  1. Dibyayan Chakraborty
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  2. L. Sunil Chandran
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  3. Sajith Padinhatteeri
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  4. Raji R. Pillai
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Correspondence to Raji R. Pillai .

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Editors and Affiliations

  1. University of Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. Universität Tübingen, Tübingen, Germany

    Michael Kaufmann

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Chakraborty, D., Chandran, L.S., Padinhatteeri, S., Pillai, R.R. (2022). s-Club Cluster Vertex Deletion on Interval and Well-Partitioned Chordal Graphs. In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_10

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  • DOI: https://doi.org/10.1007/978-3-031-15914-5_10

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