On the $(l,w)$-Domination Number of the Cube Network

Abstract

For an $n$-connected graph $G$, the $n$-wide diameter $d_n(G)$, is the minimum integer $m$ such that there are at least $n$ internally disjoint $(di)$paths of length at most $m$ between any vertices $x$ and $y$. For a given integer $l$, a subset $S$ of $V(G)$ is called an $(l,n)$-dominating set of $G$ if for any vertex $x\in V(G)–S$ there are at least $n$ internally disjoint $(di)$paths of length at most $l$ from $S$ to $z$. The minimum cardinality among all $(l,n)$-dominating sets of $G$ is called the $(l,n)$-domination number. In this paper, we obtain that the $(l,n)$-domination number of the $d$-ary cube network $C(d,n)$ is $2$ for $1\le w\le d$ and $d_w(G)–f(d,n)\le l\le d_w(G)–1$ if $d,n\ge 4$, where $f(d,n)=min\{e(\lfloor \frac{n}{2}\rceil +1),\lfloor \frac{n}{2}\rfloor e\}$.