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arXiv · 1812.03195

Counting independent sets in graphs with bounded bipartite pathwidth

Martin Dyer·Catherine Greenhill·Haiko Müller
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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.

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Martin Dyer, Catherine Greenhill, Haiko Müller. 2018-12-07. Counting independent sets in graphs with bounded bipartite pathwidth. https://arxiv.org/abs/1812.03195

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