Let k ≥ 1 be an integer. A signed Roman k-dominating function on a digraph D is a function f : V (D) −→ {−1, 1, 2} such that ∑x∈N−[v] f(x) ≥ k for every v ∈ V (D), where N−[v] consists of v and all in-neighbors of v, and every vertex u ∈ V (D) for which f(u) = −1 has an in-neighbor w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed Roman k-dominating functions on D with the pro...

When we examine the stories of women leaders in the early church—especially the stories of their actions for alleviating suffering—their teaching on the causes of suffering comes into focus: ignorance and fear which are the root of alienation, isolation, and divisions among people and from the earth. Ignorance and fear also manifest in patterns of social injustice and in organized systems of op...

An outer-independent double Roman dominating function (OIDRDF) of a graph G is h:V(G)→{0,1,2,3}</...

Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertic...

Roman remains: Extensive ruins such as those of the Roman colony of Thamugadi (Timgad) in modern-day Algeria remind us that civilisations can and do collapse. (Photo: Wikimedia Commons/PhR61.) In 100 AD, the Roman emperor Trajan founded a new colony in Northern Africa, which he called colonia Marciana Traiana Thamugadi or Thamugadi for short. Planned to cover a surface of 12 hectares with its s...