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A set S of vertices in an isolate-free graph G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number is the minimum cardinality of a semitotal dominating set of G, and is bounded below by the domination number and bounded above by the total domination number, arguably the two most important...
In this article, we use edge weighting functions on dominating sets to show that if we impose a regularity condition on a graph, then upper bounds on both the upper domination number and the upper total domination number can be greatly improved. More precisely, we prove that for if G is a k‐regular graph on n vertices, then the upper domination number of G is at most , and the upper total domination...
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