On the Distance Paired-Domination of Circulant Graphs - Wang, Haoli; Xu, Xirong; Yang, Yuansheng; Wang, Guoqing; Lu, Kai (Bulletin of the Malaysian Mathematical Sciences Society, 2011)
Let \(G=(V,E)\) be a graph without isolated vertices. A set \(D\subseteq V\) is a \(d\)-distance paired-dominating set of \(G\) if \(D\) is a \(d\)-distance dominating set of \(G\) and the induced subgraph \(G=(V,E)\) has a perfect matching. The minimum cardinality of a \(d\)-distance paired-dominating set for graph \(G\) is the \(d\)-distance paired-domination number, denoted by \(γ_p^{d}(G)\). In this paper, we study the \(d\)-distance paired-domination number of circulant graphs \(C(n;\{1,k\})\) for \(2≤ k≤ 4\). We prove that for \(k=2\), \(n≥ 5\) and \(d≥ 1\), ... for \(k=3\), \(n≥ 7\) and \(d≥ 1\), ... and for \(k=4\) and \(n≥ 9\), (i) if \(d=1\), then ... (ii) if \(d≥ 2\), then ... 2010 Mathematics Subject Classification: 05C69, 05C12. (ProQuest: ... denotes formulae omitted.)