Abstract
A set \(S\subseteq V\) is a paired-dominating set if every vertex in \(V{\setminus } S\) has at least one neighbor in S and the subgraph induced by S contains a perfect matching. The paired-domination number of a graph G, denoted by \(\gamma _{pr}(G)\), is the minimum cardinality of a paired-dominating set of G. A conjecture of Goddard and Henning says that if G is not the Petersen graph and is a connected graph of order n with minimum degree \(\delta (G)\ge 3\), then \(\gamma _{pr}(G)\le 4n/7\). In this paper, we confirm this conjecture for k-regular graphs with \(k\ge 4\).
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Supported in part by National Natural Science Foundation of China (Nos. 11371008 and 91230201) and Science and Technology Commission of Shanghai Municipality (No. 13dz2260400).
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Lu, C., Wang, C. & Wang, K. Upper Bounds for the Paired-Domination Numbers of Graphs. Graphs and Combinatorics 32, 1489–1494 (2016). https://doi.org/10.1007/s00373-015-1661-z
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DOI: https://doi.org/10.1007/s00373-015-1661-z