Independent perfect domination sets in Cayley graphs

Abstract

Abstract:In this paper, we show that a Cayley graph for an abelian group has an independent perfect domination set if and only if it is a covering graph of a complete graph. As an application, we show that the hypercube Q n has an independent perfect domination set if and only if Q n is a regular covering of the complete graph K n 1 if and only if n 2 m À 1 for some natural number m.

Total perfect codes in Cayley graphs

“…In [9] perfect 1-codes in a Cayley graph with connection set closed under conjugation were studied, yielding necessary conditions in terms of the irreducible characters of the underlying group. In [25] it was proved that a subset C of a group G closed under conjugation (or a normal subset as is called in [25]) is a perfect 1-code in a Cayley graph on G if and only if there exists a covering from the Cayley graph to a complete graph such that C is a fibre of the corresponding covering projection. Perfect 1-codes in circulants (that is, Cayley graphs on cyclic groups) were studied in [31,33].…”

Reinforcement numbers of digraphs

“…There are many important classes of networks for which it is known exactly which graphs in each class have E-sets. These classes include hypercubes, cubeconnected cycles, circulant graphs, tori, and so on [4,8,14,16]. In [8] we used these characterizations to determine the bondage number b(G), which was first introduced by Fink et al [6] as the minimum number of edges whose removal results in a graph with larger domination number.…”

(Total) Domination in Prisms

“…Knowing the existence of such codes, by the divisibility condition one immediately infers that Q n contains a perfect code if and only if n = 2 k − 1 for some k ≥ 1. Lee [15,Theorem 3] further proved that this is equivalent to the fact that Q n is a regular covering of the complete graph K n+1 . The second assertion of Theorem 2.1 is due to van Wee [23].…”