Connected components of graphs and reverse mathematics

Abstract

Reverse mathematics provides powerful techniques for analyzing the logical content of theorems. The goal of this paper is to illustrate the use of these techniques in an accessible setting, relatively free of complicated coding. All the results assess the difficulty of decomposing graphs into connected components. In Theorems 2.5 and 3.1 and their corollaries, the degree of difficulty is measured by recursion theoretic complexity. In Theorem 4.5, an induction scheme acts as another measure of difficulty. Theorem 4.5 lies outside the hierarchical axiom scheme of reverse mathematics, revealing an interesting connection between set comprehension axioms and induction schemes.