Total Domination in Circulant Graphs

A Cayley graph is a graph constructed out of a group and its generating set. In this paper, we define generalized circulant graphs and attempt to find dominating sets and independent dominating sets for the same. Actually we find the values of the domination number, independent domination number and total domination number for generalized circulant graphs. Also it is proved that certain generalized circulant graphs are excellent and in some other cases there are two excellent 2-excellent.

2009

Advanced Studies in Contemporary Mathematics, 2012

A circulant graph is a Cayley graph constructed out of a finite cyclic group Γ and a generating set A of Γ. Sharp upper bounds are obtained for the domination number, the total domination number and the connected domination number of circulant graphs and exact values for some of the parameters are determined in certain cases. Further, efficient dominating sets of some classes of circulant graphs are also obtained.

2007

Abstract A Cayley graph is a graph constructed out of a group $\ Gamma $ and its generating set $ A $. In this paper we attempt to find dominating sets in Cayley graphs constructed out of $ Z_ {n} $. Actually we find the value of domination number for $ Cay (Z_ {n}, A) $ and a minimal dominating set when $| A| $ is even and further we have proved that $ Cay (Z_ {n}, A) $ is excellent. We have also shown that $ Cay (Z_ {n}, A) $ is $2-$ excellent, when $ n= t (| A|+ 1)+ 1$ for some integer $ t, t> 0$.

Discrete Mathematics, Algorithms and Applications, 2020

Let [Formula: see text] be a simple undirected graph. [Formula: see text] is a circulant graph defined on [Formula: see text] with difference set [Formula: see text] provided two vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text]. For convenience, we use [Formula: see text] to denote such a circulant graph. A function [Formula: see text] is an integer [Formula: see text]-domination function if for each [Formula: see text], [Formula: see text] By considering all [Formula: see text]-domination functions [Formula: see text], the minimum value of [Formula: see text] is the [Formula: see text]-domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we prove that if [Formula: see text], [Formula: see text], then the integer [Formula: see text]-domination number of [Formula: see text] is [Formula: see text].

arXiv (Cornell University), 2019

Let G = (V, E) be a simple undirected graph. G is a circulant graph defined on V = Zn with difference set D ⊆ {1, 2,. .. , ⌊ n 2 ⌋} provided two vertices i and j in Zn are adjacent if and only if min{|i−j|, n−|i−j|} ∈ D. For convenience, we use G(n; D) to denote such a circulant graph. A function f : V (G) → N∪{0} is an integer {k}-domination function if for each v ∈ V (G), u∈N G [v] f (u) ≥ k. By considering all {k}-domination functions f , the minimum value of v∈V (G) f (v) is the {k}-domination number of G, denoted by γ k (G). In this paper, we prove that if D = {1, 2,. .. , t}, 1 ≤ t ≤ n−1 2 , then the integer {k}-domination number of G(n; D) is ⌈ kn 2t+1 ⌉.

2009

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G − v is less than the total domination number of G. We call these graphs t-critical. In this paper, we deter- mine the domination and the total domination number in the Circulant graphs Cnh 1, 3i , and then study -criticality and t-criticality in these graphs. Finally, we provide answers to some open questions.

2011

Abstract: A circulant graph is a Cayley graph constructed out of a finite cyclic group Γ and a generating set A is a subset of Γ. In this paper, we attempt to find upper bounds for distance-g domination, distance-g paired domination and distance-g connected domination number for circulant graphs. Exact values are also determined in certain cases. Key Words: Circulant graph, Smarandachely distance-g paired-(U, V) dominating P-set, distance-g domination, distance-g paired, total and connected domination, distance-g efficient domination.

Applied Mathematics Letters, 2012

Efficient open dominating sets in bipartite Cayley graphs are characterized in terms of covering projections. Necessary and sufficient conditions for the existence of efficient open dominating sets in certain circulant Harary graphs are given. Chains of efficient dominating sets, and of efficient open dominating sets, in families of circulant graphs are described as an application.

Communications in Algebra

A signed dominating function of graph Γ is a function g : V (Γ) −→ {−1, 1} such that u∈N [v] g(u) > 0 for each v ∈ V (Γ). The signed domination number γ S (Γ) is the minimum weight of a signed dominating function on Γ. Let G = S be a finite group such that e ∈ S = S −1. In this paper, we obtain the signed domination number of Cay(S : G) based on cardinality of S. Also we determine the classification of group G by |S| and γ S (Cay(S : G)).