In graph theory, a domatic partition of a graph
Contents
The domatic number is the maximum size of a domatic partition, that is, the maximum number of disjoint dominating sets. The graph in the figure has domatic number 3. It is easy to see that the domatic number is at least 3 because we have presented a domatic partition of size 3. To see that the domatic number is at most 3, we first review a simple upper bound.
Upper bounds
Let
The graph in the figure has minimum degree
Lower bounds
If there is no isolated vertex in the graph (that is,
The figure on the right shows a weak 2-coloring, which is also a domatic partition of size 2: the dark nodes are a dominating set, and the light nodes are another dominating set (the light nodes form a maximal independent set). See weak coloring for more information.
Computational complexity
Finding a domatic partition of size 1 is trivial: let
However, finding a maximum-size domatic partition is computationally hard. Specifically, the following decision problem, known as the domatic number problem, is NP-complete: given a graph
There is a polynomial-time approximation algorithm with a logarithmic approximation guarantee, that is, it is possible to find a domatic partition whose size is within a factor
However, under plausible complexity-theoretic assumptions, there is no polynomial-time approximation algorithm with a sub-logarithmic approximation factor. More specifically, a polynomial-time approximation algorithm for domatic partition with the approximation factor
Comparison with similar concepts
Let G = (U ∪ V, E) be a bipartite graph without isolated nodes; all edges are of the form {u, v} ∈ E with u ∈ U and v ∈ V. Then {U, V} is both a vertex 2-coloring and a domatic partition of size 2; the sets U and V are independent dominating sets. The chromatic number of G is exactly 2; there is no vertex 1-coloring. The domatic number of G is at least 2. It is possible that there is a larger domatic partition; for example, the complete bipartite graph Kn,n for any n ≥ 2 has domatic number n.