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...

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...

In this work, we study the signed Roman domination number of the join of graphs. Specially, we determine it for the join of cycles, wheels, fans, and friendship graphs.

‎Let $G$ be a finite and simple graph with vertex set $V(G)$‎. ‎A nonnegative signed total Roman dominating function (NNSTRDF) on a‎ ‎graph $G$ is a function $f:V(G)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin N(v)}f(x)ge 0$ for each‎ ‎$vin V(G)$‎, ‎where $N(v)$ is the open neighborhood of $v$‎, ‎and (ii) every vertex $u$ for which‎ ‎$f(u...

A function f : V (G) → {+1,−1} defined on the vertices of a graph G is a signed dominating function if for any vertex v the sum of function values over its closed neighborhood is at least 1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. By simply changing “{+1,−1}” in the above definition to “{+1, 0,−1}”, we can define the minus dominating f...

3 A function f : V (G) → {+1,−1} defined on the vertices of a graph G is a signed domi4 nating function if for any vertex v the sum of function values over its closed neighborhood 5 is at least one. The signed domination number γs(G) of G is the minimum weight of a 6 signed dominating function on G. By simply changing “{+1,−1}” in the above definition 7 to “{+1, 0,−1}”, we can define the minus ...

Let $D$ be a finite and simple digraph with vertex set $V(D)$. A weak signed Roman dominating function (WSRDF) on is $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the condition that $\sum_{x\in N^-[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ consists of $v$ allvertices from which arcs go into $v$. The weight WSRDF $f$ $\sum_{v\in V(D)}f(v)$. domination number $\gamma_{wsR}(D)$ minimum $D$. In...