## Trees with paired-domination number twice their domination number.(English)Zbl 1181.05070

Summary: We continue the study of paired-domination in graphs introduced by T. W. Haynes and P. J. Slater [Networks 32, No. 3, 199–206 (1998; Zbl 0997.05074)]. A paired-dominating set of a graph $$G$$ with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of $$G$$ is the minimum cardinality of a paired-dominating set of $$G$$. For $$k\geq 2$$, a $$k$$-packing in $$G$$ is a set $$S$$ of vertices of $$G$$ that are pairwise at distance greater than $$k$$ apart. The $$k$$-packing number of $$G$$ is the maximum cardinality of a $$k$$-packing in $$G$$. Haynes and Slater observed that the paired-domination number is bounded above by twice the domination number. We give a constructive characterization of the trees attaining this bound that uses labelings of the vertices. The key to our characterization is the observation that the trees with paired-domination number twice their domination number are precisely the trees with 2-packing number equal to their 3-packing number.

### MSC:

 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C05 Trees

Zbl 0997.05074