On the $(l,w)$-Domination Numbers of the Circulant Network

Abstract

For an $n$-connected graph $G$, the $n$-wide diameter $d_n(G)$ is the minimum integer $m$ such that for any two vertices $x$ and $y$ there are at least $n$ internally disjoint paths of length at most $m$ from $x$ to $y$. For a given integer $l$, a subset $S$ of $V(G)$ is called a $(l,n)$-dominating set of $G$ if for any vertex $x\in V(G)–S$ there are at least $n$ internally disjoint paths of length at most $l$ from $S$ to $x$. 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 $(l,\omega )$-domination numbers of the circulant digraph $G(d^n;\{1,d,\dots ,d^{n-1}\})$ is equal to 2 for $1\le \omega \le n$ and $d_{\omega }(G)–(g(d,n)+\delta )\le l\le d_{\omega }(G)–1$, where $g(d,n)=\text{min}\{e\lceil \frac{n}{2}\rceil –e–2,(\lfloor \frac{n}{2}\rfloor +1)(e–1)–2\}$, $\delta =0$ for $1\le \omega \le n–1$ and $\delta =1$ for $\omega =n$.