A Roman dominating function (RDF) on a graph G = (V,E) is a function f : V → {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 f is w(f) = ∑ v∈V f(v). The Roman domination number is the minimum weight of an RDF in G. It is known that for every graph G, the Roman domination number of G is bounded above...

For a finite simple edge-colored connected graph G (the coloring may not be proper), a rainbow path in G is a path without two edges colored the same; G is rainbow connected if for any two vertices of G, there is a rainbow path connecting them. Rainbow connection number, rc(G), of G is the minimum number of colors needed to color its edges such that G is rainbow connected. Chakraborty et al. (2...

In this paper, we continue the study of power domination in graphs (see SIAM J. Discrete Math. 15 (2002), 519–529; SIAM J. Discrete Math. 22 (2008), 554–567; SIAM J. Discrete Math. 23 (2009), 1382–1399). Power domination in graphs was birthed from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A set of vertices is defined to b...

In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter 2. For a tournament T of diameter 2, we show 2 ≤ − →rc(T ) ≤ 3. Furthermore, we provide a general upper bound on the rainbow k-connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of kth diameter 2 has rainbow k-co...

A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. The total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. Total domination subdivision number denoted by sdγt is the minimum number of edges that must be subdivided to increase the total domination number. Here we...

For a graph G a subsetD of the vertex set of G is a k-dominating set if every vertex not in D has at least k neighbors in D. The k-domination number γk(G) is the minimum cardinality among the k-dominating sets of G. Note that the 1-domination number γ1(G) is the usual domination number γ(G). Fink and Jacobson showed in 1985 that the inequality γk(G) ≥ γ(G) + k − 2 is valid for every connected g...

A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D. The double domination number of a graph G is the minimum cardinality of a double dominating set of G. For a graph G = (V,E), a subset D ⊆ V (G) is a 2dominating set if every vertex of V (...

Let γt(G) and γ2(G) be the total domination number and the 2domination number of a graph G, respectively. It has been shown that: γt(T ) ≤ γ2(T ) for any tree T . In this paper, we provide a constructive characterization of those trees with equal total domination number and 2-domination number.

Let G = (V (G), E(G)) be a nontrivial connected graph of order n with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ N, where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree connecting S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such t...

Let G = (V,E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the ...