Counting Connected Set Partitions of Graphs

TL;DR

This paper introduces a splitting approach to analyze the partition polynomial of graphs, explores properties of the bond lattice, discusses a bivariate extension, and proves the computational complexity of calculating it.

Contribution

It presents a novel splitting method for the partition polynomial on separating vertex sets and analyzes its properties, including the complexity of the bivariate polynomial.

Findings

Splitting approach simplifies the computation of the partition polynomial.

Properties of the bond lattice are summarized.

Computing the bivariate partition polynomial is -hard.

Abstract

Let $G=(V,E)$ be a simple undirected graph with $n$ vertices then a set partition $\pi=\{V_1, ..., V_k\}$ of the vertex set of $G$ is a connected set partition if each subgraph $G[V_j]$ induced by the blocks $V_j$ of $\pi$ is connected for $1\le j\le k$. Define $q_{i}(G)$ as the number of connected set partitions in $G$ with $i$ blocks. The partition polynomial is then $Q(G, x)=\sum_{i=0}^n q_{i}(G)x^i$. 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 $Q(G,x,y)=\sum_{i=1}^n \sum_{j=1}^m q_{ij}(G)x^iy^j$ is briefly discussed, where $q_{ij}(G)$ 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 $\sharp P$-hard.