Polytopality of maniplexes

Given an abstract polytope P, its flag graph is the edge-coloured graph whose vertices are the flags of P and whose i-edges correspond to i-adjacent flags. Flag graphs of polytopes are maniplexes. On the other hand, a maniplex need not be the flag graph of a polytope. It is natural to ask when does a maniplex is the flag graph of a polytope. In this paper we give necessary and sufficient conditions (in terms of graphs) on a maniplex to be (isomorphic to) the flag graph of a polytope. For this, given a maniplex M, we define a poset PM and determine when is PM an abstract polytope. Moreover, in such case, we show that M is isomorphic to the flag graph of PM.