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Algorithmic Aspects of Disjunctive Total Domination in Graphs

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Combinatorial Optimization and Applications (COCOA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10043))

Abstract

For a graph \(G=(V,E)\), \(D\subseteq V\) is a dominating set if every vertex in \(V\!\setminus \!D\) has a neighbor in D. If every vertex in V has to be adjacent to a vertex of D, then D is called a total dominating set of G. The (total) domination problem on G is to find a (total) dominating set D of the minimum cardinality. The (total) domination problem is well-studied. Recently, the following variant is proposed. Vertex subset D is a disjunctive total dominating set if every vertex of V is adjacent to a vertex of D or has at least two vertices in D at distance 2 from it. The disjunctive total domination problem on G is to find a disjunctive total dominating set D of the minimum cardinality. For the complexity issue, the only known result is that the disjunctive total domination problem is NP-hard on general graphs. In this paper, by using a minimum-cost flow algorithm as a subroutine, we show that the disjunctive total domination problem on trees can be solved in polynomial time. This is the first polynomial-time algorithm for the problem on a special class of graphs. Besides, we show that the problem remains NP-hard on bipartite graphs and planar graphs.

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Acknowledgement

This work was partially supported by the Ministry of Science and Technology of Taiwan, under Contract No. MOST 105-2221-E-259-018.

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Authors and Affiliations

  1. Department of Computer Science and Information Engineering, National Dong Hwa University, Hualien, 97401, Taiwan

    Chin-Fu Lin & Sheng-Lung Peng

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  1. Chin-Fu Lin
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  2. Sheng-Lung Peng
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Correspondence to Sheng-Lung Peng .

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Editors and Affiliations

  1. University of Hong Kong, Hong Kong, China

    T-H. Hubert Chan

  2. City University of Hong Kong, Hong Kong, China

    Minming Li

  3. City University of Hong Kong, Hong Kong, China

    Lusheng Wang

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Lin, CF., Peng, SL. (2016). Algorithmic Aspects of Disjunctive Total Domination in Graphs. In: Chan, TH., Li, M., Wang, L. (eds) Combinatorial Optimization and Applications. COCOA 2016. Lecture Notes in Computer Science(), vol 10043. Springer, Cham. https://doi.org/10.1007/978-3-319-48749-6_21

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  • DOI: https://doi.org/10.1007/978-3-319-48749-6_21

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-48748-9

  • Online ISBN: 978-3-319-48749-6

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