A {em weak signed Roman dominating function} (WSRDF) of a graph $G$ with vertex set $V(G)$ is defined as afunction $f:V(G)rightarrow{-1,1,2}$ having the property that $sum_{xin N[v]}f(x)ge 1$ for each $vin V(G)$, where $N[v]$ is theclosed neighborhood of $v$. The weight of a WSRDF is the sum of its function values over all vertices.The weak signed Roman domination number of $G...

A {em Roman dominating function} on a graph $G$ is a function $f:V(G)rightarrow {0,1,2}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A {em total Roman dominating function} is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has n...

Let D be a finite and simple digraph with vertex set V (D) and arc set A(D). A signed Roman dominating function (SRDF) on the digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑ x∈N−[v] f(x) ≥ 1 for each v ∈ V (D), where N −[v] consists of v and all inner neighbors of v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The w...

Let k be a positive integer and G = (V,E) be a graph of minimum degree at least k − 1. A function f : V → {−1, 1} is called a signed k-dominating function of G if ∑ u∈NG[v] f(u) ≥ k for all v ∈ V . The signed k-domination number of G is the minimum value of ∑ v∈V f(v) taken over all signed k-dominating functions of G. The signed total k-dominating function and signed total k-domination number o...

In this paper we study the signed Roman dominationnumber of the join of graphs. Specially, we determine it for thejoin of cycles, wheels, fans and friendship graphs.

a {em roman dominating function} on a graph $g = (v ,e)$ is a function $f : vlongrightarrow {0, 1, 2}$ satisfying the condition that every vertex $v$ for which $f (v) = 0$ is adjacent to at least one vertex $u$ for which $f (u) = 2$. the {em weight} of a roman dominating function is the value $w(f)=sum_{vin v}f(v)$. the roman domination number of a graph $g$, denoted by $gamma_r(g)$, equals the...