On the geodesic pre-hull number of a graph

Given a convexity space X whose structure is induced by an interval operator I , we define a parameter, called the pre-hull number of X , which measures the intrinsic non-convexity of X in terms of the number of iterations of the pre-hull operator associated with I which are necessary in the worst case to reach the canonical extension of copoints of X when they are being extended by the adjunction of an attaching point. We consider primarily the geodesic convexity structure of connected graphs in the case where the pre-hull number is at most 1, with emphasis on bipartite graphs, in particular, partial cubes.