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Connected domatic number in planar graphs. (English) Zbl 1079.05512

Summary: A dominating set in a graph \(G\) is a connected dominating set of \(G\) if it induces a connected subgraph of \(G\). The connected domatic number of \(G\) is the maximum number of pairwise disjoint, connected dominating sets in \(V(G)\). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.

MSC:

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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References:

[1] G. Chartrand and L. Lesniak: Graphs and Digraphs. Prindle, Weber & Schmidt, Boston, 1986.
[2] S. T. Hedetniemi: personal communication.
[3] S. M. Hedetniemi, S. T. Hedetniemi and R. Reynolds: Combinatorial Problems on Chessboards: II, Chapter 6. Domination in Graphs: Advanced Topics, Marcel Dekker, Inc., New York, 1997.
[4] S. T. Hedetniemi and R. Laskar: Connected domination in graphs. Graph Theory and Combinatorics, Academic Press, London-New York, 1984, pp. 209-217.
[5] E. Sampathkumar and H. B. Walikar: The connected domination number of a graph. J. Math. Phys. Sci. 13 (1979), 607-613. · Zbl 0449.05057
[6] B. Zelinka: Connected domatic number of a graph. Math. Slovaca 36 (1986), 387-392. · Zbl 0625.05042
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