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Distance-$k$ locating-dominating sets in graphs

Let $G$ be a graph with vertex set $V$, and let $k$ be a positive integer. A set $D \subseteq V$ is a \emph{distance-$k$ dominating set} of $G$ if, for each vertex $u \in V-D$, there exists a vertex $w\in D$ such that $d(u,w) \le k$, where $d(u,w)$ is the minimum number of edges linking $u$ and $w$ in $G$. Let $d_k(x, y)=\min\{d(x,y), k+1\}$. A set $R\subseteq V$ is a \emph{distance-$k$ resolving set} of $G$ if, for any pair of distinct $x,y\in V$, there exists a vertex $z\in R$ such that $d_k(x,z) \neq d_k(y,z)$. The \emph{distance-$k$ domination number} $γ_k(G)$ (\emph{distance-$k$ dimension} $\dim_k(G)$, respectively) of $G$ is the minimum cardinality of all distance-$k$ dominating sets (distance-$k$ resolving sets, respectively) of $G$. The \emph{distance-$k$ location-domination number}, $γ_L^k(G)$, of $G$ is the minimum cardinality of all sets $S\subseteq V$ such that $S$ is both a distance-$k$ dominating set and a distance-$k$ resolving set of $G$. Note that $γ_L^1(G)$ is the well-known location-domination number introduced by Slater in 1988. For any connected graph $G$ of order $n\ge 2$, we obtain the following sharp bounds: (1) $γ_k(G) \le \dim_k(G)+1$; (2) $2\leγ_k(G)+\dim_k(G) \le n$; (3) $1\le \max\{γ_k(G), \dim_k(G)\} \le γ_L^k(G) \le \min\{\dim_k(G)+1, n-1\}$. We characterize $G$ for which $γ_L^k(G)\in\{1, |V|-1\}$. We observe that $\frac{\dim_k(G)}{γ_k(G)}$ can be arbitrarily large. Moreover, for any tree $T$ of order $n\ge 2$, we show that $γ_L^k(T)\le n-ex(T)$, where $ex(T)$ denotes the number of exterior major vertices of $T$, and we characterize trees $T$ achieving equality. We also examine the effect of edge deletion on the distance-$k$ location-domination number of graphs.