Let D = (V,A) be a finite simple directed graph (shortly digraph) in which dD(v) ≥ 1 for all v ∈ V . A function f : V −→ {−1, 1} is called a signed total dominating function if ∑ u∈N−(v) f(u) ≥ 1 for each vertex v ∈ V . A set {f1, f2, . . . , fd} of signed total dominating functions on D with the property that ∑d i=1 fi(v) ≤ 1 for each v ∈ V (D), is called a signed total dominating family (of f...

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arcset $A(D)$. A twin signed total Roman dominating function (TSTRDF) on thedigraph $D$ is a function $f:V(D)rightarrow{-1,1,2}$ satisfyingthe conditions that (i) $sum_{xin N^-(v)}f(x)ge 1$ and$sum_{xin N^+(v)}f(x)ge 1$ for each $vin V(D)$, where $N^-(v)$(resp. $N^+(v)$) consists of all in-neighbors (resp.out-neighbors) of $v$, and (...

A partition of V (G), all of whose classes are dominating sets in G, is called a domatic partition of G. The maximum number of classes of a domatic partition of G is called the domatic number of G. The concept of a domatic number was introduced in [1]. More interesting results on domatically full graphs, domatically critical, domatically cocritical graphs and other domatic numbers can be found ...

‎for any integer $kgeq 1$‎, ‎a set $s$ of vertices in a graph $g=(v,e)$ is a $k$-‎tuple total dominating set of $g$ if any vertex‎ ‎of $g$ is adjacent to at least $k$ vertices in $s$‎, ‎and any vertex‎ ‎of $v-s$ is adjacent to at least $k$ vertices in $v-s$‎. ‎the minimum number of vertices of such a set‎ ‎in $g$ we call the $k$-tuple total restrained domination number of $g$‎. ‎the maximum num...

‎For any integer $kgeq 1$‎, ‎a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-‎tuple total dominating set of $G$ if any vertex‎ ‎of $G$ is adjacent to at least $k$ vertices in $S$‎, ‎and any vertex‎ ‎of $V-S$ is adjacent to at least $k$ vertices in $V-S$‎. ‎The minimum number of vertices of such a set‎ ‎in $G$ we call the $k$-tuple total restrained domination number of $G$‎. ‎The maximum num...

Let G be a finite and simple graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of distinct signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ j for each x...

Let G be a finite and simple graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑x∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ k for each x ∈ V (G), ...