A Roman dominating function of a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. Let G be a connected n-vertex graph. We prove that γR(G) ≤ 4n/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for γR(...

Let $G=(V,E)$ be a finite and simple graph of order $n$ maximumdegree $\Delta$. A signed strong total Roman dominating function ona $G$ is $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) forevery vertex $v$ $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where$N(v)$ open neighborhood (ii) every forwhich $f(v)=-1$ adjacent to at least one vertex...

The existence of a constant time algorithm for solving different domination problems on the subclass of polygraphs, rotagraphs and fasciagraphs, is shown by means of path algebras. As these graphs include products (the Cartesian, strong, direct, lexicographic) of paths and cycles, we implement the algorithm to get formulas in the case of the domination numbers, the Roman domination numbers and ...

Let G = (V,E) be a graph. A subset S ⊆ V is a dominating set of G if every vertex not in S is adjacent to a vertex in S. A set D̃ ⊆ V of a graph G = (V,E) is called an outer-connected dominating set for G if (1) D̃ is a dominating set for G, and (2) G[V \ D̃], the induced subgraph of G by V \ D̃, is connected. The minimum size among all outer-connected dominating sets of G is called the outerconnec...

‎‎Let $G=(V‎, ‎E)$ be a simple graph with vertex set $V$ and edge set $E$‎. ‎A {em mixed Roman dominating function} (MRDF) of $G$ is a function $f:Vcup Erightarrow {0,1,2}$ satisfying the condition that every element $xin Vcup E$ for which $f(x)=0$ is adjacent‎‎or incident to at least one element $yin Vcup E$ for which $f(y)=2$‎. ‎The weight of an‎‎MRDF $f$ is $sum _{xin Vcup E} f(x)$‎. ‎The mi...

Recent articles by ReVelle [20, 21] in the Johns Hopkins Magazines suggested a new variation of domination called Roman domination, see also [22] for an integer programming formulation of the problem. Since then, there have been several articles on Roman domination and its variations [2, 3, 4, 5, 6, 11, 12, 14, 15, 16, 18, 24, 23, 25]. Emperor Constantine had the requirement that an army or leg...