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 \(1 \leq w \leq d\) and \(d_w(G) – f(d, n) \leq l \leq d_w(G) – 1 \) if \(d,n\geq 4\), where \(f(d, n) = \min\{e(\left\lfloor \frac{n}{2} \right\rceil + 1), \left\lfloor \frac{n}{2} \right\rfloor e\}\).