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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.)
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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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