In this paper we initiate the study of total restrained domination in graphs. Let $G=(V,E)$ be a graph. A total restrained dominating set is a set $S\subseteq V$ where every vertex in $V-S$ is adjacent to a vertex in $S$ as well as to another vertex in $V-S$, and every vertex in $S$ is adjacent to another vertex in $S$. The total restrained domination number of $G$, denoted by ${\gamma }_r^t\left(G\right)$, is the smallest cardinality of a total restrained dominating set of $G$. First, some exact values and sharp bounds for ${\gamma }_r^t\left(G\right)$ are given in Section 2....

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...

The signed distance-$k$-domination number of a graph is a certain variant of the signed domination number. If $v$ is a vertex of a graph $G$, the open $k$-neighborhood of $v$, denoted by $N_k\left(v\right)$, is the set $N_k\left(v\right)=\{u\mid u\ne v$ and $d(u,v)\le k\}$. $N_k\left[v\right]=N_k\left(v\right)\cup \left\{v\right\}$ is the closed $k$-neighborhood of $v$. A function $fV\to \{-1,1\}$ is a signed distance-$k$-dominating function of $G$, if for every vertex $v\in V$, $f\left(N_k\left[v\right]\right)={\sum }_{u\in N_k\left[v\right]}f\left(u\right)\ge 1$. The signed distance-$k$-domination number, denoted by ${\gamma }_{k,s}\left(G\right)$, is the minimum weight of a signed distance-$k$-dominating function on $G$. The values of ${\gamma }_{2,s}\left(G\right)$ are found for graphs...