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. (shrink)

A recursive graph is a graph whose vertex and edge sets are recursive. A highly recursive graph is a recursive graph that also has the following property: one can recursively determine the neighbors of a vertex. Both of these have been studied in the literature. We consider an intermediary notion: Let A be a set. An A-recursive graph is a recursive graph that also has the following property: one can recursively-in-A determine the neighbors of a vertex. We show that, if (...) A is r.e. and not recursive, then there exists A-recursive graphs that are 2-colorable but not recursively k-colorable for any k. This is false for highly-recursive graphs but true for recursive graphs. Hence A-recursive graphs are closer in spirit to recursive graphs than to highly recursive graphs. (shrink)

A Π01 class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of the members of a class P. Given an effective enumeration {Pe:e < ω} of the Π01 classes, the index set I for a certain property is the set of indices e such that Pe has the property. For example, the index set of binary Π01 classes of positive measure is Σ02 complete. (...) Various notions of boundedness are discussed and classified. For example, the index set of the recursively bounded classes is Σ03 complete and the index set of the recursively bounded classes which have infinitely many recursive members is Π04 complete. Consideration of the Cantor-Bendixson derivative leads to index sets in the transfinite levels of the hyperarithmetic hierarchy. (shrink)

Several papers (e.g. [7, 23, 42]) have recently sought to give general frameworks for online structures and algorithms ([4]), and seeking to connect, if only by analogy, online and computable structure theory. These initiatives build on earlier work on online colouring and other combinatorial algorithms by Bean [10], Kierstead, Trotter et al. [48, 54, 57] and others, as we discuss below. In this paper we will look at such frameworks and illustrate them with examples from the first author’s MSc Thesis (...) ([58]). In this thesis, Askes looks at online, adversarial online and strongly online algorithms and structures. Additionally, we prove some new theorems about online algorithms on graphs of bounded pathwidth as well as some new results on punctually 1-decidable structures. (shrink)