  It is an open problem to show that in two-dimensional first-passage
percolation, the sequence of finite geodesics from any point to $(n,0)$ has a
limit in $n$. In this paper, we consider this question for first-passage
percolation on a wide class of subgraphs of $\mathbb {Z}^2$: those whose vertex
set is infinite and connected with an infinite connected complement. This
includes, for instance, slit planes, half-planes and sectors. Writing $x_n$ for
the sequence of boundary vertices, we show that the sequence of geodesics from
any point to $x_n$ has an almost sure limit assuming only existence of finite
geodesics. For all passage-time configurations, we show existence of a limiting
Busemann function. Specializing to the case of the half-plane, we prove that
the limiting geodesic graph has one topological end; that is, all its infinite
geodesics coalesce, and there are no backward infinite paths. To do this, we
prove in the Appendix existence of geodesics for all product measures in our
domains and remove the moment assumption of the Wehr-Woo theorem on absence of
bigeodesics in the half-plane.
