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Domatic partitions of computable graphs


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.

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  1. Bean D.: Effective coloration. J. Symb. Logic 41(2), 469–480 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  2. Diestel R.: Graph Theory. 2nd edn. Springer, New York (2000)

    Google Scholar 

  3. Gasarch, W.: A survey of recursive combinatorics. In: Ershov, Goncharov, Marek, Nerode, Remmel (eds.) Handbook of Recursive Mathematics, vol. 2, pp. 1041–1176. Elsevier, Amsterdam (1998)

  4. Gasarch W., Hirst J.: Reverse mathematics and recursive graph theory. MLQ Math. Log. Q. 44(4), 465–473 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Harel D.: Hamiltonian paths in infinite graphs. Israel J. Math. 76, 317–336 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hirst J., Lempp S.: Infinite versions of some problems from finite complexity theory. Notre Dame J. Formal Logic 37(4), 545–553 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Riege, T.: The Domatic Number Problem: Boolean Hierarchy Completeness and Exact Exponential-Time Algorithms, Ph.D. Dissertation, Heinrich-Heine-Universität Düsseldorf, Düsseldorf (2006)

  8. Rogers H. Jr: Theory of Recursive Functions and Effective Computability. The MIT Press, Cambridge (1987)

    Google Scholar 

  9. Simpson S. G.: Subsystems of Second Order Arithmetic. Springer, New York (1998)

    Google Scholar 

  10. Soare R.I.: Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic. Springer, New York (1987)

    Google Scholar 

  11. Zelinka B.: Domatic number and degrees of vertices of a graph. Mathematica Slovaca 33(2), 145–147 (1983)

    MATH  MathSciNet  Google Scholar 

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Correspondence to Tyler Markkanen.

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Jura, M., Levin, O. & Markkanen, T. Domatic partitions of computable graphs. Arch. Math. Logic 53, 137–155 (2014).

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  • Computability theory
  • Computable graph theory
  • Domatic number
  • Dominating set
  • Reverse mathematics

Mathematics Subject Classification (2000)

  • 03D45
  • 05C63
  • 05C69