Bounds
Let G be a graph with n ≥ 1 vertices and let Δ be the maximum degree of the graph. The following bounds on γ(G) are known (Haynes, Hedetniemi & Slater 1998a, Chapter 2):
- One vertex can dominate at most Δ other vertices; therefore γ(G) ≥ n/(1 + Δ).
- The set of all vertices is a dominating set in any graph; therefore γ(G) ≤ n.
If there are no isolated vertices in G, then there are two disjoint dominating sets in G; see domatic partition for details. Therefore in any graph without isolated vertices it holds that γ(G) ≤ n/2.
Read more about this topic: Dominating Set
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