Arc and twin arc domination in Cayley digraphs

Abstract

Let G be a finite group and let S be a generating subset of G. The Cayley digraph Cay (G, S) is the digraph with node set as the elements of G and there is an arc from u to us whenever u ϵ G and s ϵ S. An arc e1 in a digraph D out arc dominates itself as well as all arcs ei. Such that {e1, ei} is a directed path of length 2 in D. While e1 in arc dominates both itself and all arcs ej such that {ei, e1} is a directed path of length 2 in D. The arc domination number is the minimum cardinality of an out arc dominating set of D denoted by γ′(D). A set of edges of D is twin arc dominating set if every edge of D is out arc dominated by some edge of S and in arc dominated by some edge of S. The minimum Cardinality of a twin arc dominating set is the twin arc domination number denoted by γ* (D) of D. This paper discusses the arc domination and twin arc domination of Cayley digraphs and attempts to find bounds for the arc domination number of the same class of graphs.