Abstract
A set $$S\subseteq V$$S⊆V is a paired-dominating set if every vertex in $$V{\setminus } S$$V\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 $$\gamma _{pr}(G)$$?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 $$\delta (G)\ge 3$$?(G)?3, then $$\gamma _{pr}(G)\le 4n/7$$?pr(G)≤4n/7. In this paper, we confirm this conjecture for k-regular graphs with $$k\ge 4$$k?4.
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