Counting Connected Set Partitions of Graphs

Let G = (V,E) be a simple undirected graph with n vertices then a set partition π = {V1, . . . , Vk} of the vertex set of G is a connected set partition if each subgraph G[Vj ] induced by the blocks Vj of π is connected for 1 ≤ j ≤ k. Define qi(G) as the number of connected set partitions in G with i blocks. The partition polynomial is Q(G,x) = ∑n i=0 qi(G)x . This paper presents a splitting approach to the partition polynomial on a separating vertex set X in G and summarizes some properties of the bond lattice. Furthermore the bivariate partition polynomial Q(G,x, y) = ∑n i=1 ∑m j=1 qij(G)x y is briefly discussed, where qij(G) counts the number of connected set partitions with i blocks and j intra block edges. Finally the complexity for the bivariate partition polynomial is proven to be ♯P -hard.