On the Distance Paired-Domination of Circulant Graphs

• Published in: Bulletin of the Malaysian Mathematical Sciences Society Vol. 34; no. 1
• Main Authors: Wang, Haoli, Xu, Xirong, Yang, Yuansheng, Wang, Guoqing, Lu, Kai
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.01.2011
• Subjects: Mathematics
• ISSN: 0126-6705; 2180-4206

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 \(\gamma_p^{d}(G)\). In this paper, we study the \(d\)-distance paired-domination number of circulant graphs \(C(n;\{1,k\})\) for \(2\leq k\leq 4\). We prove that for \(k=2\), \(n\geq 5\) and \(d\geq 1\), ... for \(k=3\), \(n\geq 7\) and \(d\geq 1\), ... and for \(k=4\) and \(n\geq 9\), (i) if \(d=1\), then ... (ii) if \(d\geq 2\), then ... 2010 Mathematics Subject Classification: 05C69, 05C12. (ProQuest: ... denotes formulae omitted.)