  In this work we propose a novel method to calculate mean first-passage times
(MFPTs) for random walks on graphs, based on a dimensionality reduction
technique for Markov State Models, known as local-equilibrium (LE). We show
that for a broad class of graphs, which includes trees, LE coarse-graining
preserves the MFPTs between certain nodes, upon making a suitable choice of the
coarse-grained states (or clusters). We prove that this relation is exact for
graphs that can be coarse-grained into a one-dimensional lattice where each
cluster connects to the lattice only through a single node of the original
graph. A side result of the proof generalises the well-known essential edge
lemma (EEL), which is valid for reversible random walks, to irreversible
walkers. Such a generalised EEL leads to explicit formulae for the MFPTs
between certain nodes in this class of graphs. For graphs that do not fall in
this class, the generalised EEL provides useful approximations if the graph
allows a one-dimensional coarse-grained representation and the clusters are
sparsely interconnected.
  We first demonstrate our method for the simple random walk on the $c$-ary
tree, then we consider other graph structures and more general random walks,
including irreversible random walks.
