Related papers: Hardness and Algorithmic Results for Roman \{3\}-D…

A Roman dominating function of a graph $G=(V,E)$ is a labeling $f: V \rightarrow{} \{0 ,1, 2\}$ such that for each vertex $u \in V$ with $f(u) = 0$, there exists a vertex $v \in N(u)$ with $f(v) =2$. A Roman dominating function $f$ is a…

Given a graph $G = (V, E)$, a signed Roman dominating function is a function $f: V \rightarrow \{-1, 1, 2\}$ such that for every vertex $u \in V$: $\sum_{v \in N[u]} f(v) \geq 1$ and for every vertex $u \in V$ with $f(u) = -1$, there exists…

Consider a graph $G = (V, E)$ and a function $f: V \rightarrow \{0, 1, 2\}$. A vertex $u$ with $f(u)=0$ is defined as \emph{undefended} by $f$ if it lacks adjacency to any vertex with a positive $f$-value. The function $f$ is said to be a…

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…

Given a graph $G=(V,E)$, a function $f:V\to \{0,1,2\}$ is said to be a \emph{Roman Dominating function} if for every $v\in V$ with $f(v)=0$, there exists a vertex $u\in N(v)$ such that $f(u)=2$. A Roman Dominating function $f$ is said to be…

Given a graph $G$ with vertex set $V(G)$, a mapping $h : V(G) \rightarrow \lbrace 0, 1, 2, 3, 4, 5 \rbrace$ is called a quadruple Roman dominating function (4RDF) for $G$ if it holds the following. Every vertex $x$ such that $h(x)\in…

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 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…

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 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…

In the Roman domination problem, an undirected simple graph $G(V,E)$ is given. The objective of Roman domination problem is to find a function $f:V\rightarrow {\{0,1,2\}}$ such that for any vertex $v\in V$ with $f(v)=0$ must be adjacent to…

Let $G=(V(G),E(G))$ be a simple graph. A restrained double Roman dominating function (RDRD-function) of $G$ is a function $f: V(G) \rightarrow \{0,1,2,3\}$ satisfying the following properties: if $f(v)=0$, then the vertex $v$ has at least…

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…

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…

For a graph $G= (V, E)$, a Roman dominating function is a map $f : V \rightarrow \{0, 1, 2\}$ satisfies the property that if $f(v) = 0$, then $v$ must have adjacent to at least one vertex $u$ such that $f(u)= 2$. The weight of a Roman…

For a graph $G=(V,E)$, a Roman $\{2\}$-dominating function (R2DF)$f:V\rightarrow \{0,1,2\}$ has the property that for every vertex $v\in V$ with $f(v)=0$, either there exists a neighbor $u\in N(v)$, with $f(u)=2$, or at least two neighbors…

For a positive integer $k$, a $\{k\}$-Roman dominating function of a graph $G = (V,E)$ is a function $f\colon V \rightarrow \{0,1,\ldots,k\}$ satisfying $f (N(v)) \geq k$ for each vertex $v\in V$ with $f (v) = 0$. Every graph $G$ satisfies…

Given a graph $G$ with vertex set $V$, $f : V \rightarrow \{0, 1, 2\}$ is a \emph{Roman $\{2\}$-dominating function} (or \emph{italian dominating function}) of $G$ if for every vertex $v\in V$ with $f(v) =0$, either there exists a vertex…

Domination in graphs is a widely studied field, where many different definitions have been introduced in the last years to respond to different network requirements. This paper presents a new dominating parameter based on the well-known…

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 $f : V (G)\rightarrow \{0, 1,\ldots , \left\lceil \frac{\Delta}{2}\right\rceil+ 1\}$…