Upper Bounds for the Paired-Domination Numbers of Graphs

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 $$\gamma _{pr}(G)$$γpr(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 $$\delta (G)\ge 3$$δ(G)≥3, then $$\gamma _{pr}(G)\le 4n/7$$γpr(G)≤4n/7. In this paper, we confirm this conjecture for k-regular graphs with $$k\ge 4$$k≥4.