Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by γc(G) + 2, where γc(G) is th...

The domination number γ(G), the independent domination number ι(G), the connected domination number γc(G), and the paired domination number γp(G) of a graph G (without isolated vertices, if necessary) are related by the simple inequalities γ(G) ≤ ι(G), γ(G) ≤ γc(G), and γ(G) ≤ γp(G). Very little is known about the graphs that satisfy one of these inequalities with equality. I.E. Zverovich and V...

A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The Roman domination number of G, γR(G), is the minimum weight of a Roman dominating function on G. In this paper, we...

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) ...

Let G be a graph of order n ≥ 2 and n1, n2, .., nk be integers such that 1 ≤ n1 ≤ n2 ≤ .. ≤ nk and n1 + n2 + .. + nk = n. Let for i = 1, .., k: Ai ⊆ Kni where Km is the set of all pairwise non-isomorphic graphs of order m, m = 1, 2, ... In this paper we study when for a domination related parameter μ (such as domination number, independent domination number and acyclic domination number) is ful...

A subset D of ( ) V G is called an equitable dominating set if for every ( ) v V G D   there exists a vertex u D  such that ( ) uv E G  and | ( ) ( ) | 1 deg u deg v   . A subset D of ( ) V G is called an equitable independent set if for any , u D v   ( ) e N u for all { } v D u   . The concept of equi independent equitable domination is a combination of these two important concepts. ...

Let G be a simple graph of order n, maximum degree Δ 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 best possible upper and lower bounds, as functions of n, Δ and δ, for the independent domination number of graphs G wi...