TOTAL DOMINATION VERSUS PAIRED-DOMINATION IN REGULAR GRAPHS

A subset S of vertices of a graph G is a dominating set of G if every vertex not in S has a neighbor in S, while S is a total dominating set of G if every vertex has a neighbor in S. If S is a dominating set with the additional property that the subgraph induced by S contains a perfect matching, then S is a paired-dominating set. The domination number, denoted gamma(G), is the minimum cardinality of a dominating set of G, while the minimum cardinalities of a total dominating set and paired-dominating set are the total domination number, gamma t(G), and the paired-domination number, gamma(pr)(G), respectively. For k >= 2, let G be a connected k-regular graph. It is known [Schaudt, Total domination versus paired domination, Discuss. Math. Graph Theory 32 (2012) 435447] that gamma(pr)(G)/gamma t(G) <= (2k)/(k+1). In the special case when k = 2, we observe that gamma(pr)(G)/gamma(t)(G) <= 4/3, with equality if and only if G congruent to C5. When k = 3, we show that gamma(pr)(G)/gamma(t)(G) <= 3/2, with equality if and only if G is the Petersen graph. More generally for k >= 2, if G has girth at least 5 and satisfies gamma(pr)(G)/gamma(t)(G) = (2k)/(k + 1), then we show that G is a diameter-2 Moore graph. As a consequence of this result, we prove that for k >= 2 and k not equal 57, if G has girth at least 5, then gamma(pr)(G)/gamma(t)(G) <= (2k)/(k +1), with equality if and only if k = 2 and G congruent to C5 or k = 3 and G is the Petersen graph.