  Relation Gelfand-Tsetlin $\mathfrak{gl}_n$-modules were introduced in
[FRZ19], and are determined by some special directed graphs and Gelfand-Tsetlin
characters. In this work we constructed polyhedra associated with the class of
relation modules, which includes as a particular case, any classical
Gelfand-Tsetlin polytope. Following the ideas presented in [LM04] we give a
characterization of $d$-faces of the associated polyhedra in terms of a matrix
related to the corresponding graph.
