Article in volume
Authors:
Title:
End super dominating sets in graphs
PDFSource:
Discussiones Mathematicae Graph Theory 45(1) (2025) 21-47
Received: 2023-02-17 , Revised: 2023-08-19 , Accepted: 2023-08-21 , Available online: 2023-09-25 , https://doi.org/10.7151/dmgt.2519
Abstract:
Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset
$S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one
vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by
$\gamma(G)$, is the domination number of $G$. A super dominating set is a
dominating set $S$ with the additional property that every vertex in
$V \setminus S$ has a neighbor in $S$ that is adjacent to no other vertex in
$V \setminus S$. Moreover if every vertex in $V \setminus S$ has degree at
least $2$, then $S$ is an end super dominating set. The end super domination
number is the minimum cardinality of an end super dominating set. We give
applications of end super dominating sets as main servers and temporary servers
of networks. We determine the exact value of the end super domination number
for specific classes of graphs, and we count the number of end super dominating
sets in these graphs. Tight upper bounds on the end super domination number are
established, where the graph is modified by vertex (edge) removal and contraction.
Keywords:
domination number, end super dominating set, end super domination number, networks, generalization
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