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A function π : V → {1,. .. , k} is a broadcast coloring of order k if π(u) = π(v) implies that the distance between u and v is more than π(u). The minimum order of a broadcast coloring is called the broadcast chromatic number of G, and is denoted χ b (G). In this paper we introduce this coloring and study its properties. In particular, we explore the… (More)

A Roman dominating function 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 a Roman dominating function is the value f (V) = u∈V f (u). The minimum weight of a Roman dominating function on a graph G is called the… (More)

A set S is an offensive alliance if for every vertex v in its boundary N (S) − S it holds that the majority of vertices in v's closed neighbourhood are in S. The offensive alliance number is the minimum cardi-nality of an offensive alliance. In this paper we explore the bounds on the offensive alliance and the strong offensive alliance numbers (where a… (More)

We say that a function f : V → {0, 1,. .. , diam(G)} is a broadcast if for every vertex v ∈ V , f (v) ≤ e(v), where diam(G) denotes the diameter of G and e(v) denotes the eccentricity of v. The cost of a broadcast is the value f (V) = Σ v∈V f (v). In this paper we introduce and study the minimum and maximum costs of several types of broadcasts in graphs,… (More)