Counting connected sets and connected partitions of a graph

This paper concerns two related enumeration problems on vertex labeled graphs. Given such a graph G, we investigate the number C(G) of connected subsets of the vertex set and the number P (G) of connected partitions of the vertex set. By connected we mean that the induced subgraphs are connected. The numbers C(G) and P (G) can be regarded as the (connected) graph analogs of the number of subsets and the number of set partitions, respectively, of an n-element set.