Lower bounds for subgraph detection in the CONGEST model

In the subgraph-freeness problem, we are given a constant-sized graph H, and wish to determine whether the network graph contains H as a subgraph or not. Until now, the only lower bounds on subgraph-freeness known for the CONGEST model were for cycles of length greater than 3; here we extend and generalize the cycle lower bound, and obtain polynomial lower bounds for subgraph-freeness in the CONGEST model for two classes of subgraphs. The first class contains any graph obtained by starting from a 2-connected graph H for which we already know a lower bound, and replacing the vertices of H by arbitrary connected graphs. We show that the lower bound on H carries over to the new graph. The second class is constructed by starting from a cycle C_k of length k ≥ 4, and constructing a graph H from C_k by replacing each edge {i, (i + 1) mod k} of the cycle with a connected graph H_i, subject to some constraints on the graphs H₀, ⋯, H_k-1. In this case we obtain a polynomial lower bound for the new graph H, depending on the size of the shortest cycle in H passing through the vertices of the original k-cycle.