The curve graph 
of a connected orientable
surface 
is the graph whose vertices
are isotopy classes of essential closed
curves on 
without self-intersections. Here a curve is essential when it is neither homotopic
to a point nor homotopic to a component of the boundary
of 
. Two vertices are adjacent when the
corresponding classes have disjoint representatives (Harvey 1981, Masur and Minsky
1999).
Equivalently, the curve graph is the 1-skeleton of the simplicial complex whose simplices are finite sets of classes having pairwise disjoint representatives.
For exceptional low-complexity surfaces on which distinct essential classes exist but cannot be disjoint, adjacency is instead defined using representatives with the smallest possible positive number of intersections (Masur and Minsky 1999).
The curve graph should not be confused with a curvy graph, which is a graph whose rectilinear crossing number exceeds its graph crossing number.
