Domination parameters in random graphs G(n, p), where p is a fixed real number in (0, 1), are investigated. We show that with probability tending to 1 as n → ∞, the total and independent domination numbers concentrate on the domination number of G(n, p).

Abstract Given a graph G and subset of vertices $$D\subseteq V(G)$$ D ⊆ V ( G ) , the external neighbourhood D is defined as $$N_e(D)=\{u\in V(G){\setminus } D:\, N(u)\cap D\ne \varnothing \}$$ <mml:msu...

Let G be a simple graph of order n, maximum degree ∆ and minimum degree δ ≥ 2. The independent domination number i(G) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. The girth g(G) is the minimum length of a cycle in G. We establish sharp upper and lower bounds, as functions of n, ∆ and δ, for the independent domination number of graphs G with g(G) ...

In this paper, the bottleneck dominating set problem and one of its variants, the bottleneck independent dominating set problem, are considered. Let G(V, E, W) denote a graph with n-vertex-set V and m-edge-set E, where each vertex v is associated with a real cost W(v). Given any subset V′ of V, the bottleneck cost of V′ is defined as max{W(x)  x ∈ V′}. The major task involves identifying a dom...

Ve consider the independence, domination and independent domination numbers of graphs obtained from the moves of queens on chessboards drawn on the torus, and determine exact values for each of these parameters in infinitely many cases.