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...

<abstract><p>Let $ G be a graph with vertex set V(G) $. A function f:V(G)\rightarrow \{0, 1, 2\} is Roman dominating on if every v\in for which f(v) = 0 adjacent to at least one u\in such that f(u) 2 The domination number of the minimum weight \omega(f) \sum_{x\in V(G)}f(x) among all functions f In this article we study direct product graphs and rooted graphs. Specifically, give sev...