Containment Properties of Product and Power Graphs

In this paper we study containment properties of graphs in relation with the Cartesian product operation. We show that the isomorphism of two Cartesian powers Gr and Hr implies the isomorphism of G and H, while Gr ⊆ Hr does not imply G ⊆ H, even for the special cases when G and H are connected or have the same number of nodes. Then, we find a simple sufficient condition under which the containment of products implies the containment of the factors: if Πin=1 Gii ⊆ Πin=1 Hj, where all graphs Gi are connected and no graph Hj has 4-cycles, then each Gi is a subgraph of a different graph Hj. Hence, if G is connected and H has no 4-cycles, then Gr ⊆ Hr implies G ⊆ H. These results can be used to derive embedding results for interconnection networks for parallel architectures.