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Independent Domination in Bipartite Cubic Graphs

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Abstract

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by i(G), is the minimum cardinality of an independent dominating set. In this paper, we study the following conjecture posed by Goddard and Henning (Discrete Math. 313:839–854, 2013): If \(G\not \cong K_{3,3}\) is a connected, cubic, bipartite graph on n vertices, then \(i(G) \le \frac{4}{11}n\). Henning et al. (Discrete Appl. Math. 162:399–403, 2014) prove the conjecture when the girth is at least 6. In this paper we strengthen this result by proving the conjecture when the graph has no subgraph isomorphic to \(K_{2,3}\).

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Acknowledgements

The authors express their sincere thanks to the referees for their meticulous and thorough reading of the paper.

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Authors and Affiliations

  1. Institute of Discrete Mathematics and Algebra, TU Bergakademie Freiberg, 09596, Freiberg, Germany

    Christoph Brause

  2. Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa

    Christoph Brause & Michael A. Henning

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  1. Christoph Brause
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  2. Michael A. Henning
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Correspondence to Christoph Brause.

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M. A. Henning: Research supported in part by the University of Johannesburg.

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Brause, C., Henning, M.A. Independent Domination in Bipartite Cubic Graphs. Graphs and Combinatorics 35, 881–912 (2019). https://doi.org/10.1007/s00373-019-02043-0

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  • DOI: https://doi.org/10.1007/s00373-019-02043-0

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