Efficient dominating sets in Cayley graphs (original) (raw)

Efficient k-Distance Dominating Set in Cayley Graphs

Proceedings of the National Academy of Sciences, India. Section A, Physical sciences, 2018

The efficient k-distance domination number c k e ðGÞ is the minimum among the cardinalities of efficient k-distance dominating sets of G. Upper bound in terms of order and maximum degree of an independent set S of vertices in a graph G ¼ ðV; EÞ called an efficient k-distance dominating set if every vertex in V À S be distance k from exactly one vertex in S has been presented. Sufficient conditions for 3-regular Cayley graphs to have disjoint efficient k-distance dominating sets are given. Characterization of the 3-regular connected circulant graphs that admit an efficient k-distance dominating set for k ¼ 2; 3 has been obtained.

Efficient open domination in Cayley graphs

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.

Dominating sets in Cayley graphs on $ Z_ {n} $

2007

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

On the signed domination number of some Cayley graphs

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)).

Efficient Domination in Bi-Cayley Graphs

2010

Abstract: A Cayley graph is constructed out of a group Γ and its generating set X and it is denoted by С (Γ, X). A Smarandachely n-Cayley graph is defined to be G= ZC (Γ, X), where V (G)= Γ× Zn and E (G)={((x, 0),(y, 1)) a,((x, 1),(y, 2)) a,···,((x, n− 2),(y, n− 1)) a: x, y∈ Γ, a∈ X such that y= x∗ a}. Particularly, a Smarandachely 2-Cayley graph is called as a Bi-Cayley graph, denoted by BС (Γ, X).

Worst-case efficient dominating sets in digraphs

2012

Let 1< n∈. Worst-case efficient dominating sets in digraphs are conceived so that their presence in certain strong digraphs S⃗T⃗_n corresponds to that of efficient dominating sets in star graphs ST_n: The fact that the star graphs ST_n form a so-called dense segmental neighborly E-chain is reflected in a corresponding fact for the digraphs S⃗T⃗_n. Related chains of graphs and open problems are presented as well.

Dominating Functions and Cayley Graphs

2019

Let G = (V,E) be a graph. A function f : V → [0, 1] is called a dominating function if ∑ u∈N [v] f(u) ≥ 1 for every vertex v ∈ V . Let f and g be any two functions from V to [0, 1], f 6= g. We say f is less than g and we write f < g if f(u) ≤ g(u) for all u ∈ V . A dominating function f of G is said to be minimal dominating function if whenever g < f , g is not a dominating function of G. In this Paper we study these functions for certain classes of graphs which includes Quadratic Residue Cayley graphs in particular.