For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices in a connected graph $G$ and a vertex $v$ of $G$, the code of $v$ with respect to $W$ is the $k$-vector $$c_W\left(v\right)=(d(v,w_1),d(v,w_2),\cdots ,d(v,w_k)).$$ The set $W$ is an independent resolving set for $G$ if (1) $W$ is independent in $G$ and (2) distinct vertices have distinct codes with respect to $W$. The cardinality of a minimum independent resolving set in $G$ is the independent resolving number $\mathrm{i}r\left(G\right)$. We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs $G$ of order...

P. Kristiansen, S. M. Hedetniemi, and S. T. Hedetniemi, in Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004), 157–177, and T. W. Haynes, S. T. Hedetniemi, and M. A. Henning, in Global defensive alliances in graphs, Electron. J. Combin. 10 (2003), introduced the defensive alliance number $a\left(G\right)$, strong defensive alliance number $\widehat{a}\left(G\right)$, and global defensive alliance number ${\gamma }_a\left(G\right)$. In this paper, we consider relationships between these parameters and the domination number $\gamma \left(G\right)$. For any positive...

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.

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...