Let SG be the category of simplicial groups and let HAk be the category of connected Hopf algebras over a field k. A graph product of m objects C1, …, Cm in C (with C=SG or HAk) with respect to a graph G with m vertices, is defined by the free product of these objects subject to relations, given by their commutators, from the edges of G. In this talk we show that the bar construction (in the sense of Eilenberg-MacLane) of such a graph product, which is an object in the category of simplicial sets when C=SG or graded chain complexes over k when C=HAk, can be reduced to a much smaller object, namely the polyhedral product of BC1, …, BCm with respect to the flag complex K by filling all triangles in G. We will also show several applications of this result.