Connected domination number in circulant graphs

Abstract

Let (Γ, *) be a finite group and e be its identity. Let A be a generating set of Γ such that e / ∈ A and a −1 ∈ A for all a ∈ A. Then the Cayley graph is defined by G = (V (G), E(G)), where V (G) = Γ and E(G) = {(x, x * a) |x ∈ V (G), a ∈ A}, denoted by Cay(Γ, A). Circulant graphs are special case of Cayley graphs when Γ = (Z n , ⊕ n), where ⊕ n is the operation addition modulo n. In this paper, for the Circulant graphs Cir(n, A), where A = {1, 2,. .. k, n − 1, n − 2,. .. , n − k} and k ≤ n−1 2 , the connected domination number has been obtained.

2-Tuple total domination number in circulant graphs

“…By the definition of A, D is a independent vertex subset of V (G). We prove that D is an efficient 2-dominating set of G. The next result is a special case of the above theorem when a = 1, which was proved in [5].…”

“…. , n − x} and x ≤ n−1 2 [5]. In this paper, we continue the study of a variation of the domination theme, namely that of perfect 2-tuple total domination.…”

Variants of the domination number for flower snarks

“…For total domination, results are known for some grid graphs G(n, m) (for n ≤ 6) [15,21], Knödel graphs [27], and various graphs from chemistry [26]. For connected domination, results are known for trees [33], circulant graphs [35], Centipede graphs [38], and various graphs from chemistry [26]. For weak Roman domination, results are known for various graphs based on chessboards [30], Cartesian products involving complete graphs [37], and Helm graphs and Web graphs [23].…”

Variants of the Domination Number for Flower Snarks

“…For total domination, results are known for some grid graphs G(n, m) (for n ≤ 6) [13,20], Knödel graphs [19], and various graphs from chemistry [25]. For connected domination, results are known for trees [31], circulant graphs [33], Centipede graphs [36], and various graphs from chemistry [25]. For weak Roman domination, results are known for various graphs based on chessboards [29], Cartesian products involving complete graphs [35], and Helm graphs and Web graphs [22].…”