  A consistent path system in a graph $G$ is an intersection-closed collection
of paths, with exactly one path between any two vertices in $G$. We call $G$
metrizable if every consistent path system in it is the system of geodesic
paths defined by assigning some positive lengths to its edges. We show that
metrizable graphs are, in essence, subdivisions of a small family of basic
graphs with additional compliant edges. In particular, we show that every
metrizable graph with 11 vertices or more is outerplanar plus one vertex.
