  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.
