Graph-truncations of $3$-polytopes

In this paper we study the operation of cutting off edges of a simple $3$-polytope $P$ along the graph $\mathrm{\Gamma }$. We give the criterion when the resulting polytope is simple and when it is flag. As a corollary we prove the analog of Eberhard’s theorem about the realization of polygon vectors of simple $3$-polytopes for flag polytopes.