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Domination in bipartite graphs and in their complements. (English) Zbl 1021.05074
Summary: The domatic number of a graph $$G$$ and of its complement $$\overline G$$ were studied by J. E. Dunbar, T. W. Haynes and M. A. Henning. They suggested four open problems. We solve the following ones:
Characterize the bipartite graphs $$G$$ with $$d(G) = d(\overline G)$$. Further, we present a partial solution to the problem: Is it true that if $$G$$ is a graph satisfying $$d(G) = d(\overline G)$$, then $$\gamma (G) = \gamma (\overline G)$$? Finally, we prove an existence theorem concerning the total domatic number of a graph and of its complement.
##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
##### Keywords:
bipartite graph; complement of a graph; domatic number
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##### References:
 [1] E. J. Cockayne and S. T. Hedetniemi: Towards the theory of domination in graphs. Networks 7 (1977), 247-261. · Zbl 0384.05051 [2] E. J. Cockayne, R. M. Dawes and S. T. Hedetniemi: Total domination in graphs. Networks 10 (1980), 211-219. · Zbl 0447.05039 [3] J. E. Dunbar, T. W. Haynes and M. A. Henning: The domatic number of a graph and its complement. Congr. Numer. 8126 (1997), 53-63. · Zbl 0902.05039
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