On an equivalence in discrete extremal problems

We introduce some equivalence relations on graphs and posets and prove that they are closed under the cartesian product operation. These relations concern the edge-isoperimetric problem on graphs and the shadow minimization problems on posets. For a long time these problems have been considered quite independently. We present close connections between them. In particular we show that a number of known results concerning the edge-isoperimetric problem for concrete families of graphs are direct consequences of the Macauleyness of appropriate posets.