Before revision:   We show that the graph H_n is connected for odd n, but has exponentially many small connected components for even n, which we characterize and count via Catalan and generalized Narayana numbers.
After revision:  The graph \mathcal{G We show that the graph H_n is connected for odd n, but has exponentially many small connected components for even n, which we characterize and count via Catalan and generalized Narayana numbers.