The well known types of routes in graphs and directed graphs, such as walks, trails, paths, and induced paths, are characterized using axioms on vertex sequences. Thus non-graphic characterizations of the various types of routes are obtained.
A transit function $R$ on a set $V$ is a function $R\:V\times V\rightarrow 2^{V}$ satisfying the axioms $u\in R(u,v)$, $R(u,v)=R(v,u)$ and $R(u,u)=\lbrace u\rbrace $, for all $u,v \in V$. The all-paths transit function of a connected graph is characterized by transit axioms.