# Sandra Mitchell Hedetniemi

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 χb(G). In this paper we introduce this coloring and study its properties. In particular, we explore the(More)
• Discrete Mathematics
• 2004
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)
Consider placing a guard on each vertex of a dominating set S0 of a graph. If for every vertex v / ∈ S0, there is a corresponding guard at an adjacent vertex u for which the resulting set S1 = S0 − {u}∪{v} is dominating, then we say that S0 is 1-secure. It is eternally 1-secure if for any sequence v1, v2, . . . , vk of vertices, there exists a sequence of(More)
• Discrete Applied Mathematics
• 2006
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 ) = Σv∈V f(v). In this paper we introduce and study the minimum and maximum costs of several types of broadcasts in graphs,(More)