# Total Domination Versus Paired-Domination in Regular Graphs

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## 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, γt(G), and the paired-domination number, γpr(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 γpr(G)/γt(G) ≤ (2k)/(k+1). In the special case when k = 2, we observe that γpr(G)/γt(G) ≤ 4/3, with equality if and only if GC5. When k = 3, we show that γpr(G)/γt(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 γpr(G)/γt(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 ≠ 57, if G has girth at least 5, then γpr(G)/γt(G) ≤ (2k)/(k +1), with equality if and only if k = 2 and GC5 or k = 3 and G is the Petersen graph.

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# Discussiones Mathematicae Graph Theory

## The Journal of University of Zielona Góra

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researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs

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