A three-valued function $fV\to \{-1,0,1\}$ defined on the vertices of a graph $G=(V,E)$ is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every $v\in V$, $f\left(N\right(v\left)\right)\ge 1$, where $N\left(v\right)$ consists of every vertex adjacent to $v$. The weight of an MTDF is $f\left(V\right)=\sum f\left(v\right)$, over all vertices $v\in V$. The minus total domination number of a graph $G$, denoted ${\gamma }_t^{-}\left(G\right)$, equals the minimum weight of an MTDF of $G$. In this paper, we discuss some properties of minus total domination on a graph...

We initiate the study of signed majority total domination in graphs. Let $G=(V,E)$ be a simple graph. For any real valued function $fV\to ℝ$ and $S\subseteq V$, let $f\left(S\right)={\sum }_{v\in S}f\left(v\right)$. A signed majority total dominating function is a function $fV\to \{-1,1\}$ such that $f\left(N\right(v\left)\right)\ge 1$ for at least a half of the vertices $v\in V$. The signed majority total domination number of a graph $G$ is ${\gamma }_{\mathrm{m}aj}^{\mathrm{t}}\left(G\right)=min\{f\left(V\right)\mid f$ is a signed majority total dominating function on $G\}$. We research some properties of the signed majority total domination number of a graph $G$ and obtain a few lower bounds of ${\gamma }_{\mathrm{m}aj}^{\mathrm{t}}\left(G\right)$. ...

Let $G$ be a graph with vertex set $V\left(G\right)$, and let $k\ge 1$ be an integer. A subset $D\subseteq V\left(G\right)$ is called a if every vertex $v\in V\left(G\right)-D$ has at least $k$ neighbors in $D$. The $k$-domination number ${\gamma }_k\left(G\right)$ of $G$ is the minimum cardinality of a $k$-dominating set in $G$. If $G$ is a graph with minimum degree $\delta \left(G\right)\ge k+1$, then we prove that $${\gamma }_{k+1}\left(G\right)\le \frac{\left|V\left(G\right)\right|+{\gamma }_k\left(G\right)}{2}.$$ In addition, we present a characterization of a special class of graphs attaining equality in this inequality.

Let $G$ be a simple graph. A subset $S\subseteq V$ is a dominating set of $G$, if for any vertex $v\in V-S$ there exists a vertex $u\in S$ such that $uv\in E\left(G\right)$. The domination number, denoted by $\gamma \left(G\right)$, is the minimum cardinality of a dominating set. In this paper we prove that if $G$ is a 4-regular graph with order $n$, then $\gamma \left(G\right)\le \frac{4}{11}n$.

A set D of vertices in a graph G = (V,E) is a dominating set of G if every vertex in V-D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set. We define the cobondage number $b_c\left(G\right)$ of G to be the minimum cardinality among the sets of edges X ⊆ P₂(V) - E, where P₂(V) = X ⊆ V:|X| = 2 such that γ(G+X) < γ(G). In this paper, the exact values of bc(G) for some standard graphs are found and some bounds are obtained. Also, a Nordhaus-Gaddum...