Related papers: Roman domination number of zero-divisor graphs ove…

A Roman dominating function on a graph $G=(V,E)$ is a function $f: V\to \{0,1,2\}$ 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 dominating function…

The Roman dominating function on a graph $G=(V,E)$ is a function $f: V\rightarrow\{0,1,2\}$ such that each vertex $x$ with $f(x)=0$ is adjacent to at least one vertex $y$ with $f(y)=2$. The value $f(G)=\sum\limits_{u\in V(G)} f(u)$ is…

A Roman dominating function on a graph $G=(V,E)$ is a function $f:V\rightarrow\{0,1,2\}$ such that every vertex $v\in V$ with $f(v)=0$ has at least one neighbor $u\in V$ with $f(u)=2$. The weight of a Roman dominating function is the value…

A Roman dominating function for a (non-weighted) graph $G=(V,E)$, is a function $f:V\rightarrow \{0,1,2\}$ such that every vertex $u\in V$ with $f(u)=0$ has at least {one} neighbor $v\in V$ such that $f(v)=2$. The minimum weight $\sum_{v\in…

A Roman dominating function (RD-function) on a graph $G = (V(G), E(G))$ is a labeling $f : V(G) \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The weight $f(V(G))$ of a RD-function $f$ on $G$…

Given a graph $G=(V,E)$, the dominating number of a graph is the minimum size of a vertex set, $V' \subseteq V$, so that every vertex in the graph is either in $V'$ or is adjacent to a vertex in $V'$. A Roman Dominating function of $G$ is…

A quasi-total Roman dominating function on a graph $G=(V, E)$ is a function $f : V \rightarrow \{0,1,2\}$ satisfying the following: - every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) =2$, and - if…

A map $f : V \rightarrow \{0, 1, 2\}$ is a Roman dominating function on a graph $G=(V,E)$ if for every vertex $v\in V$ with $f(v) = 0$, there exists a vertex $u$, adjacent to $v$, such that $f(u) = 2$. The weight of a Roman dominating…

A $Roman\ domination\ function$ on a graph $G=(V, E)$ is a function $f:V(G)\rightarrow\{0,1,2\}$ 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…

A Roman domination function on a graph G is a function $r:V(G)\to \{0,1,2\}$ satisfying the condition that every vertex $u$ for which $r(u)=0$ is adjacent to at least one vertex $v$ for which $r(v)=2$. The weight of a Roman function is the…

Given a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2\}$ is a total Roman $\{2\}$-dominating function if: (1) every vertex $v\in V$ for which $f(v)=0$ satisfies that $\sum_{u\in N(v)}f(u)\geq 2$, where $N(v)$ represents the open…

A set $S$ of vertices of a graph $G$ is a dominating set for $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. The minimum cardinality of a dominating set for $G$ is called the domination number of $G$.…

For a graph $G = (V, E)$, a Roman dominating function $f : V \rightarrow \{0, 1, 2\}$ has the property that every vertex $v \in V $with $f (v) = 0$ has a neighbor $u$ with $f (u) = 2$. The weight of a Roman dominating function $f$ is the…

The middle graph $M(G)$ of a graph $G$ is the graph obtained by subdividing each edge of $G$ exactly once and joining all these newly introduced vertices of adjacent edges of $G$. A perfect Roman dominating function on a graph $G$ is a…

A Roman dominating function (RDF) on a graph $G = (V, E)$ is a labeling $f : V \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V) = \Sigma_{v\in V} f(v)$. The…

Consider a finite and simple graph $G=(V,E)$ with maximum degree $\Delta$. A strong Roman dominating function over the graph $G$ is understood as a map $f : V (G)\rightarrow \{0, 1,\ldots , \left\lceil \frac{\Delta}{2}\right\rceil+ 1\}$…

A total Roman dominating function on a graph $G$ is a function $% f:V(G)\rightarrow \{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ is adjacent to some vertex $u$ with $f(u)=2$, and the subgraph of $G$ induced by the set of all vertices…

Based on the history that the Emperor Constantine decreed that any undefended place (with no legions) of the Roman Empire must be protected by a "stronger" neighbor place (having two legions), a graph theoretical model called Roman…

For a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2\}$ is called Roman dominating function (RDF) if for any vertex $v$ with $f(v)=0$, there is at least one vertex $w$ in its neighborhood with $f(w)=2$. The weight of an RDF $f$ of $G$…

A double Roman dominating function of a graph $G$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ having the property that for each vertex $v$ with $f(v)=0$, there exists $u\in N(v)$ with $f(u)=3$, or there are $u,w\in N(v)$ with…