  We consider the first-passage percolation problem on effectively
one-dimensional graphs with vertex set {1,...,n}\times{0,1} and
translation-invariant edge-structure. For three of six non-trivial cases we
obtain exact expressions for the asymptotic percolation rate \chi\ by solving
certain recursive distributional equations and invoking results from ergodic
theory to identify \chi\ as the expected asymptotic one-step growth of the
first-passage time from (0,0) to (n,0).
