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Edge weighting functions on semitotal dominating sets. (English) Zbl 1441.05171
Summary: 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 domination parameters. The upper semitotal domination number, $$\Gamma_{t2}(G)$$, of $$G$$ is the maximum cardinality of a minimal semitotal dominating set in $$G$$. If $$G$$ is a connected graph with minimum degree $$\delta \geq 1$$ and of order $$n \geq \delta + 2$$, then we show that $$\Gamma_{t2}(G) \leq n - \delta$$, and that this bound is sharp for every fixed $$\delta \geq 1$$. Using edge weighting functions on semitotal dominating sets we show that if we impose a regularity condition on a graph, then this upper bound on the upper semitotal domination number can be greatly improved. We prove that if $$G$$ is a 2-regular graph on $$n$$ vertices with no $$K_3$$-component, then $$\Gamma_{t2}(G) \leq \frac{4}{7}n$$, with equality if and only if every component of $$G$$ is a cycle of length congruent to zero modulo 7. For $$k \geq 3$$, we prove that if $$G$$ is a $$k$$-regular graph on $$n$$ vertices, then $$\Gamma_{t2}(G) \leq \frac{1}{2}n$$, and we characterize the infinite families of graphs that achieve equality in this bound.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
##### Keywords:
upper semitotal domination; regular graphs
Full Text:
##### References:
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