# Counting independent sets in graphs with bounded bipartite pathwidth

## TL;DR

This paper demonstrates that for graphs with bounded bipartite pathwidth, a simple Markov chain can efficiently sample independent sets nearly uniformly, extending previous work to broader graph classes.

## Contribution

The paper introduces bipartite pathwidth as a new parameter and proves efficient sampling of independent sets for graphs with bounded bipartite pathwidth, including claw-free graphs.

## Key findings

- Efficient polynomial-time sampling of independent sets in graphs with bounded bipartite pathwidth.
- Extension of results to graphs with polynomially-bounded vertex weights.
- Identification of classes like claw-free graphs with bounded bipartite pathwidth.

## Abstract

We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity $\lambda$, can be viewed as a strong generalisation of Jerrum and Sinclair's work on approximately counting matchings, that is, independent sets in line graphs. The class of graphs with bounded bipartite pathwidth includes claw-free graphs, which generalise line graphs. We consider two further generalisations of claw-free graphs and prove that these classes have bounded bipartite pathwidth. We also show how to extend all our results to polynomially-bounded vertex weights.