Abstract
A sequence of vertices \((v_1,\, \dots , \,v_k)\) of a graph G is called a dominating closed neighborhood sequence if \(\{v_1,\, \dots , \,v_k\}\) is a dominating set of G and \(N[v_i]\nsubseteq \cup _{j=1}^{i-1} N[v_j]\) for every i. A graph G is said to be \(k-\)uniform if all dominating closed neighborhood sequences in the graph have equal length k. Brešar et al. (Discrete Math 336:22–36, 2014) characterized k-uniform graphs with \(k\le 3\). In this article we extend their work by giving a complete characterization of all k-uniform graphs with \(k\ge 4\).
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Acknowledgements
I would like to thank Didem Gözüpek and Martin Milanič for making me aware of the problem of characterization of k-uniform graphs. Also, I would like to thank anonymous referees for their helpful comments and for pointing out the references and observations in Section 3 which greatly reduced the proof of Corollary 2.
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Erey, A. Uniform Length Dominating Sequence Graphs. Graphs and Combinatorics 36, 1819–1825 (2020). https://doi.org/10.1007/s00373-020-02221-5
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DOI: https://doi.org/10.1007/s00373-020-02221-5