Abstract
A dominating set is called a global dominating set if it is a dominating set of a graph G and its complement G¯. Here we explore the possibility to relate the domination number of graph G and the global domination number of the larger graph obtained from G by means of various graph operations. In this paper we consider the following problem: Does the global domination number remain invariant under any graph operations? We present an affirmative answer to this problem and establish several results.
Highlights
The domination in graphs is one of the concepts in graph theory which has attracted many researchers to work on it
We present here an affirmative answer to this question for the graphs obtained by various graph operations on Cn and Wn
If Cn is a graph obtained by duplication of an edge in Cn (n ≠ 3, n ≠ 5) by an edge every γ-set of Cn is a global dominating set of Cn and γ(Cn ) = γg(Cn ) = ⌈n/3⌉
Summary
The domination in graphs is one of the concepts in graph theory which has attracted many researchers to work on it. If Cn is a graph obtained by duplication of an edge in Cn (n ≠ 3, n ≠ 5) by an edge every γ-set of Cn is a global dominating set of Cn and γ(Cn ) = γg(Cn ) = ⌈n/3⌉. S being a γ-set of G is a global dominating set of G with minimum cardinality n which implies that γ(G) = γg(G) = n as required.
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