Geodesic
Paired-domination
Discussiones Mathematicae. Graph Theory, Tome 18 (1998) no. 1, pp. 63-72

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We are interested in dominating sets (of vertices) with the additional property that the vertices in the dominating set can be paired or matched via existing edges in the graph. This could model the situation of guards or police where each has a partner or backup. This paper will focus on those graphs in which the number of matched pairs of a minimum dominating set of this type equals the size of some maximal matching in the graph. In particular, we characterize the leafless graphs of girth seven or more of this type.
Keywords: domination, paired-domination, matching 
Fitzpatrick, S.; Hartnell, B. Paired-domination. Discussiones Mathematicae. Graph Theory, Tome 18 (1998) no. 1, pp. 63-72. http://geodesic.mathdoc.fr/item/DMGT_1998_18_1_a4/
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[1] T.W. Haynes and P.J. Slater, Paired-domination in graphs, submitted for publication.

[2] T.W. Haynes and P.J. Slater, Paired-domination and the paired-domatic number, Congressus Numerantium 109 (1995) 65-72.