On $d$-distance $m$-tuple ($\ell, r$)-domination in graphs

In this article, we study the $d$-distance $m$-tuple ($\ell, r$)-domination problem. Given a simple undirected graph $G=(V, E)$, and positive integers $d, m, \ell$ and $r$, a subset $V' \subseteq V$ is said to be a $d$-distance $m$-tuple ($\ell, r$)-dominating set if it satisfies the following conditions: (i) each vertex $v \in V$ is $d$-distance dominated by at least $m$ vertices in $V'$, and (ii) each $r$ size subset $U$ of $V$ is $d$-distance dominated by at least $\ell$ vertices in $V'$. Here, a vertex $v$ is $d$-distance dominated by another vertex $u$ means the shortest path distance between $u$ and $v$ is at most $d$ in $G$. A set $U$ is $d$-distance dominated by a set of $\ell$ vertices means size of the union of the $d$-distance neighborhood of all vertices of $U$ in $V'$ is at least $\ell$. The objective of the $d$-distance $m$-tuple ($\ell, r$)-domination problem is to find a minimum size subset $V' \subseteq V$ satisfying the above two conditions. We prove that the problem of deciding whether a graph $G$ has (i) a 1-distance $m$-tuple ($\ell, r$)-dominating set for each fixed value of $m, \ell$, and $r$, and (ii) a $d$-distance $m$-tuple ($\ell, 2$)-dominating set for each fixed value of $d (> 1), m$, and $\ell$ of cardinality at most $k$ (here $k$ is a positive integer) are NP-complete. We also prove that for any $\varepsilon>0$, the 1-distance $m$-tuple $(\ell, r)$-domination problem and the $d$-distance $m$-tuple $(\ell,2)$-domination problem cannot be approximated within a factor of $(\frac{1}{2}- \varepsilon)\ln |V|$ and $(\frac{1}{4}- \varepsilon)\ln |V|$, respectively, unless $P = NP$.