In his article published in 1999, Ian Stewart discussed a strategy of Emperor Constantine for defending the Roman Empire. Motivated by this article, Cockayne et al.(2004) introduced the notion of Roman domination in graphs. Let G = (V,E) be a graph. A Roman dominating function of G is a function f : V → {0, 1, 2} such that every vertex v for which f(v) = 0 has a neighbor u with f(u) = 2. The we...

Mystery cults were the religious cults within the Greco-Roman world that were secretive in nature but offered their followers the notion of life after death. Mystery cults that threatened Roman political life and the state religion faced suppression and persecution. The mystery cults of Bacchus and Isis were the cults that the Roman government attacked the most. The persecution happened for ent...

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

Résumé. Très populaire et très efficace pour l’estimation de paramètres d’un modèle de mélange, l’algorithme EM présente l’inconvénient majeur de converger parfois lentement. Son application sur des tableaux de grande taille devient ainsi irréalisable. Afin de remédier à ce problème, plusieurs méthodes ont été proposées. Nous présentons ici le comportement d’une méthode connue, LEM, et d’une va...

A Roman domination function on a graph G = (V,E) is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman domination function f is the value f(V (G)) = ∑ u∈V (G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G, denoted by...

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 γR(G) of G is the minimum weight of a Roman dominating function on G. In this paper, we s...

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that for every vertex with label 0, there is a vertex with label 2 at distance at most k from each other. The weight of a k-distance Roman dominating function f is the value ω(f) = ∑ v∈V f(v). The k-distance Roman domination number ...

The establishment of a standing Roman army during the reign of Augustus resulted in an increased demand for military doctors. The knowledge about the Roman military medicine comes primarily from the excavations at the valetudinaria. Medical instruments, medicine containers and remains of medicinal plants found there indicate that the Roman army strived to provide unwell legionaries with excelle...