Characterization of curve map graphs

In this paper we provide a characterization of curve map graphs as defined by Gavril and Schönheim, and also give a recognition algorithm for them.A curve map graph is the dual of a map obtained by placing a finite number of two-way infinite Jordan curves in the Euclidean plane in such a way that each curve divides the plane into two regions, no two curves intersect in more than one point, and any two curves which intersect at a point cross at that point.Our method is based on Gavril and Schönheim's approach, but corrects several difficulties in their characterization.