## 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.)

### Keywords:

connected dominating set; connected domatic number; planar
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### References:

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