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Proper 3-Dominating Sets in Graphs

Author

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  • Danmei Chen

    (College of Sciences, Shanghai Institute of Technology, Shanghai 201418, China)

  • Shuangjie Cai

    (College of Sciences, Shanghai Institute of Technology, Shanghai 201418, China)

Abstract

A dominating set is a classic concept that is widely used in road safety, disaster rescue operations, and chemical graphs. In this paper, we introduce a variation of the dominating set: the proper 3-dominating set. For a proper 3-dominating set D of graph G , any vertex outside D is adjacent to at least three vertices inside D , and there exists one vertex outside D that is adjacent to three vertices inside D . For graph G , the proper 3-domination number is the minimum cardinality among all proper 3-dominating sets of G . We find that a graph with minimum degree at least 3 or one for which there exists a subgraph with some characteristic always contains a proper 3-dominating set. Further, we find that when certain conditions are met, some graph products, such as the joint product, strong product, lexicographic product, and corona product of two graphs, have a proper 3-dominating set. Moreover, we discover the bounds of the proper 3-domination number. For some special graphs, we get their proper 3-domination numbers.

Suggested Citation

  • Danmei Chen & Shuangjie Cai, 2025. "Proper 3-Dominating Sets in Graphs," Mathematics, MDPI, vol. 13(12), pages 1-13, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:12:p:1960-:d:1678841
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    References listed on IDEAS

    as
    1. Y. Caro & Y. Roditty, 1990. "A note on the k -domination number of a graph," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 13, pages 1-2, January.
    2. Bermudo, Sergio & Higuita, Robinson A. & Rada, Juan, 2020. "Domination in hexagonal chains," Applied Mathematics and Computation, Elsevier, vol. 369(C).
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