Total domination versus paired-domination in regular graphs

Abstract

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 γ(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, \gt(G), and the paired-domination number, \gp(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) 435--447] that \gpr(G)/γt(G) \le (2 k)/(k + 1). In the special case when k = 2, we observe that \gpr(G)/γt(G) \le 4/3, with equality if and only if G \cong C5. When k = 3, we show that \gpr(G)/γt(G) \le 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 \gpr(G)/γt(G) = (2 k)/(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 \ne 57, if G has girth at least 5, then \gpr(G)/γt(G) \le (2 k)/(k + 1), with equality if and only if k=2 and G \cong C5 or k = 3 and G is the Petersen graph.