On the geodesic pre-hull number of a graph

Given a convexity space XX whose structure is induced by an interval operator II, we define a parameter, called the pre-hull number of XX, which measures the intrinsic non-convexity of XX in terms of the number of iterations of the pre-hull operator associated with II which are necessary in the worst case to reach the canonical extension of copoints of XX 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.