On an equivalence in discrete extremal problems Original Research Article

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.