Archive for Mathematical Logic

, Volume 53, Issue 1–2, pp 137–155

# Domatic partitions of computable graphs

Article

## Abstract

Given a graph G, we say that a subset D of the vertex set V is a dominating set if it is near all the vertices, in that every vertex outside of D is adjacent to a vertex in D. A domatic k-partition of G is a partition of V into k dominating sets. In this paper, we will consider issues of computability related to domatic partitions of computable graphs. Our investigation will center on answering two types of questions for the case when k = 3. First, if domatic 3-partitions exist in a computable graph, how complicated can they be? Second, a decision problem: given a graph, how difficult is it to decide whether it has a domatic 3-partition? We will completely classify this decision problem for highly computable graphs, locally finite computable graphs, and computable graphs in general. Specifically, we show the decision problems for these kinds of graphs to be $${\Pi^{0}_{1}}$$ -, $${\Pi^{0}_{2}}$$ -, and $${\Sigma^{1}_{1}}$$ -complete, respectively.

### Keywords

Computability theory Computable graph theory Domatic number Dominating set Reverse mathematics

### Mathematics Subject Classification (2000)

03D45 05C63 05C69

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