Abstract
In this paper, sharp upper bounds for the domination number, total domination number and connected domination number for the Cayley graph G = Cay(D2n, Ω) constructed on the finite dihedral group D2n, and a specified generating set Ω of D2n. Further efficient dominating sets in G = Cay(D2n, Ω) are also obtained. More specifically, it is proved that some of the proper subgroups of D2n are efficient domination sets. Using this, an E-chain of Cayley graphs on the dihedral group is also constructed.
Highlights
Introduction and NotationDesign of interconnection networks is an important integral part of any parallel processing of distributed system
An excellent survey of interconnection networks based on Cayley graphs can be found in [1]
The concept of domination for Cayley graphs has been studied by various authors [2,3,4,5,6,7]
Summary
Design of interconnection networks is an important integral part of any parallel processing of distributed system. Serra [3] obtained efficient dominating sets for Cayley graphs constructed on a class of groups containing permutation groups. G is the minimum cardinality of a dominating set in G and the corresponding dominating set is called a -set. The total domination number t(G) equals the minimum cardinality among all the total dominating sets in G and the corresponding total dominating set is called a t-set. The connected domination number c(G) of a graph G equals the minimum cardinality of a connected dominating set in G and a corresponding connected dominating set is called a c-set. We obtain upper bounds for domination number, total domination number and connected domination number in a Cayley graph G Cay D2n , Ω constructed on the dihedral group D2n , for n 3 and a generating set Ω. G Cay D2n , Ω , where Ω is a generating set of D2n
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