Reverse mathematics and rank functions for directed graphs

Abstract.

A rank function for a directed graph G assigns elements of a well ordering to the vertices of G in a fashion that preserves the order induced by the edges. While topological sortings require a one-to-one matching of vertices and elements of the ordering, rank functions frequently must assign several vertices the same value. Theorems stating basic properties of rank functions vary significantly in logical strength. Using the techniques of reverse mathematics, we present results that require the subsystems ${\ensuremath{\vec{RCA}_0}}$, ${{\vec{ACA}_0}}$, ${{\vec{ATR}_0}}$, and ${{\vec{\Pi^1_1 - CA}_0}}$.