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Algorithms and Complexity of s-Club Cluster Vertex Deletion

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Combinatorial Algorithms (IWOCA 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12757))

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Abstract

An s-club is a graph which has diameter at most s. Let G be a graph. A set of vertices \(D\subseteq V(G)\) is an s-club deleting (s -CD) set if each connected component of \(G-D\) is an s-club. In the s -Club Cluster Vertex Deletion (s -CVD) problem, the goal is to find an s-CD set with minimum cardinality. When \(s=1\), the s -CVD is equivalent to the well-studied Cluster Vertex Deletion problem. On the negative side, we show that unless the Unique Games Conjecture is false, there is no \((2-\epsilon )\)-algorithm for 2-CVD on split graphs, for any \(\epsilon > 0\). This contrast the polynomial-time solvability of Cluster Vertex Deletion on split graphs. We show that for each \(s\ge 2\), s-CVD is NP-hard on bounded degree planar bipartite graphs and APX-hard on bounded degree bipartite graphs. On the positive side, we give a polynomial-time algorithm to solve s-CVD on trapezoid graphs, for each \(s\ge 1\).

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Notes

  1. 1.

    Let H be a graph and E(H) be the edge set of H. A transitive orientation of H is an orientation of the edges in E(H) such that if \((a,b),(b,c) \in E(G)\) and are oriented from a to b and b to c respectively then \((a,c) \in E(H)\) and is oriented from a to c.

  2. 2.

    Such representation of trapezoid graphs are possible, see [9].

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Authors and Affiliations

  1. Indian Institute of Science, Bangalore, India

    Dibyayan Chakraborty, L. Sunil Chandran, Sajith Padinhatteeri & Raji R. Pillai

  2. BITS-Pilani, Hyderabad, India

    Dibyayan Chakraborty, L. Sunil Chandran, Sajith Padinhatteeri & Raji R. Pillai

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  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 Ottawa, Ottawa, ON, Canada

    Paola Flocchini

  2. University of Ottawa, Ottawa, ON, Canada

    Lucia Moura

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Chakraborty, D., Chandran, L.S., Padinhatteeri, S., Pillai, R.R. (2021). Algorithms and Complexity of s-Club Cluster Vertex Deletion. In: Flocchini, P., Moura, L. (eds) Combinatorial Algorithms. IWOCA 2021. Lecture Notes in Computer Science(), vol 12757. Springer, Cham. https://doi.org/10.1007/978-3-030-79987-8_11

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  • DOI: https://doi.org/10.1007/978-3-030-79987-8_11

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