Distance $k$-resolving domination number of graphs.

For $k \geq 1$, in a graph $G=(V,E)$, a set of vertices $D$ is a distance $k$-dominating set of $G$, if any vertex in $V$ not in $D$ is at distance at most $k$ from some vertex in $D$. An ordered set of vertices $W=\{w_1,w_2,\ldots,w_r\}$ is a resolving set of $G$, if for any two distinct vertices $x$ and $y$ in $V\setminus W$, there exists $1\leq i\leq r$, such that $d_G(x,w_i)\neq d_G(x,w_i)$. The minimum cardinality of such set is the metric dimension of the graph $G$, and denoted by $dim(G)$. In this paper, we introduce the distance $k$-resolving dominating set, which is a subset of $V$ that is both a distance $k$-dominating set and a resolving set of $G$. The minimum cardinality of any distance $k$-resolving dominating set of $G$, is called the distance $k$-resolving domination number, and denoted by $\gamma^r_{k}(G)$. First, we give sharp bounds for $\gamma^r_{k}(G)$ with the metric dimension $dim(G)$ and the distance $k$-domination number $\gamma_k(G)$, and we present the relationship between $\gamma^r_{k}(G)$ and $dim(G)$. Then, for any $k \geq 2$, we characterize the connected graphs having the distance $k$-resolving domination number respectively equal to $1$, $n-2$, or $n-1$. Later, we provide an upper bound for $\gamma^r_{k}(G)$ involving the order $n$ of $G$ and its diameter, and we bound the order $n$ in terms of $\gamma^r_{k}(G)$, while showing graphs achieving these bounds for any $k\geq 2$. We also establish Nordhaus-Gaddum bounds for the distance $k$-resolving domination number, for all $k\geq 2$.