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Edge Criticality in Graph Domination

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Abstract

A vertex subset \(D\subseteq V\) of a graph \(G=(V,E)\) is a dominating set of G if each vertex of G is a member of D or is adjacent to a member of D. The cardinality of a smallest dominating set of G is called the domination number of G and a nonempty graph G is q-critical if q is the smallest number of arbitrary edges of G whose removal from G necessarily increases the domination number of the resulting graph. The classes of q-critical graphs of order n are characterised in this paper for all admissible combinations of values of n and q.

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Acknowledgments

The author is indebted to Dr Anton de Villiers for producing the graphics in this paper. The anonymous reviewers are also thanked for their suggestions which have improved the quality of the paper.

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  1. Department of Industrial Engineering, Stellenbosch University, Private Bag X1, Matieland, 7602, South Africa

    Jan H van Vuuren

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  1. Jan H van Vuuren
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Correspondence to Jan H van Vuuren.

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van Vuuren, J.H. Edge Criticality in Graph Domination. Graphs and Combinatorics 32, 801–811 (2016). https://doi.org/10.1007/s00373-015-1607-5

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  • DOI: https://doi.org/10.1007/s00373-015-1607-5

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