The face-width, also called representativity, of a graph embedding in a closed surface other than the sphere is the smallest possible number of intersection points between the embedded graph and a closed curve in the surface that cannot be continuously contracted to a point (Mohar 1997). Face-width therefore measures how densely a particular embedding meets the noncontractible curves in the surface, and can vary between embeddings of the same graph.
Fiedler et al. (1995) showed that a nonplanar graph with a projective plane graph
embedding of face-width 
has graph genus 
, where 
is the floor function.
Such a graph is therefore a toroidal graph iff 
or 
. For sufficiently large 
, the toroidal crossing
number of the graph also grows at least quadratically with 
(Gitler et al. 2008).
