‎‎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...

A k-rainbow dominating function of a graph G is a map f from V (G) to the set of all subsets of {1, 2, . . . , k} such that {1, . . . , k} = ⋃ u∈N(v) f(u) whenever v is a vertex with f(v) = ∅. The k-rainbow domination number of G is the invariant γrk(G), which is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by a k-rainbow dominating function. We focu...

Let G = (V, E) be a graph. A function f : V → {−1,+1} defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. The signed total domination number of G, γ t (G), is the minimum weight of a signed total dominating function of G. In this paper, we study the signed total domination number of generalized Petersen...

Abstract. Let G = (V,E) be a simple graph. A function f : V → {−1, 1} is called an inverse signed total dominating function if the sum of its function values over any open neighborhood is at most zero. The inverse signed total domination number of G, denoted by γ0 st(G), equals to the maximum weight of an inverse signed total dominating function of G. In this paper, we establish upper bounds on...

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...

The in-neighborhood, I(v), of a vertex v in a digraph D=(V, A) is v together with the set of all vertices sending an arc to v, i.e., vertices u such that (u, v) # A. A subset of V is called dominating if it meets I(v) for every v # V. (To avoid confusion, it must be noted that some authors require in the definition meeting every out-neighborhood.) A set of vertices is called independent if no t...

Let G = (V,E) be a simple and undirected graph. For some integer k > 1, a set D ⊆ V is said to be a k-dominating set in G if every vertex v of G outside D has at least k neighbors in D. Furthermore, for some real number α with 0 < α 6 1, a set D ⊆ V is called an α-dominating set in G if every vertex v of G outside D has at least α×dv neighbors in D, where dv is the degree of v in G. The cardina...