Proper 3-Dominating Sets in Graphs

Danmei Chen () and Shuangjie Cai
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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

Mathematics, 2025, vol. 13, issue 12, 1-13

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.

Keywords: domination number; dominating set; 3-dominating set; proper 3-dominating set (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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