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On the Algorithmic Complexity of Double Vertex-Edge Domination in Graphs

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WALCOM: Algorithms and Computation (WALCOM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11355))

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Abstract

Let \(G=(V,E)\) be a simple graph. A vertex \(v\in V\) ve-dominates every edge uv incident to v, as well as every edge adjacent to these incident edges. A set \(D\subseteq V\) is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of D. The double vertex-edge dominating problem is to find a minimum double vertex-edge dominating set of G. In this paper, we show that minimum double vertex-edge dominating problem is NP-complete for chordal graphs. A linear time algorithm to find the minimum double vertex-edge dominating set for proper interval graphs is proposed. We also show that the minimum double vertex-edge domination problem cannot be approximated within \((1-\varepsilon )\ln |V|\) for any \(\varepsilon >0\) unless NP\(\subseteq \) DTIME\((|V|^{O(\log \log |V|)})\). Finally, we prove that the minimum double vertex-edge domination problem is APX-complete for graphs with maximum degree 5.

Supported by National Board for Higher Mathematics, Mumbai, India. (Ref No.: NBHM/R.P.1/2015/Fresh/168).

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Authors and Affiliations

  1. Department of Mathematics, School of Humanities and Sciences, SASTRA Deemed University, Thanjavur, 613 401, India

    Y. B. Venkatakrishnan & H. Naresh Kumar

Authors
  1. Y. B. Venkatakrishnan
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  2. H. Naresh Kumar
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Correspondence to H. Naresh Kumar .

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Editors and Affiliations

  1. Indian Institute of Technology Guwahati, Guwahati, India

    Gautam K. Das

  2. Indian Institute of Technology Guwahati, Guwahati, India

    Partha S. Mandal

  3. Indian Statistical Institute, Kolkata, India

    Krishnendu Mukhopadhyaya

  4. Gunma University, Kiryu, Japan

    Shin-ichi Nakano

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Venkatakrishnan, Y.B., Naresh Kumar, H. (2019). On the Algorithmic Complexity of Double Vertex-Edge Domination in Graphs. In: Das, G., Mandal, P., Mukhopadhyaya, K., Nakano, Si. (eds) WALCOM: Algorithms and Computation. WALCOM 2019. Lecture Notes in Computer Science(), vol 11355. Springer, Cham. https://doi.org/10.1007/978-3-030-10564-8_15

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  • DOI: https://doi.org/10.1007/978-3-030-10564-8_15

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10563-1

  • Online ISBN: 978-3-030-10564-8

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