Let P be a set of n vertices in the plane and S a set of non-crossing
line segments between vertices in P, called constraints. Two vertices are
visible if the straight line segment connecting them does not properly
intersect any constraints. The constrained Θm-graph is constructed by
partitioning the plane around each vertex into m disjoint cones, each with
aperture θ=2π/m, and adding an edge to the `closest' visible vertex
in each cone. We consider how to route on the constrained Θ6-graph. We
first show that no deterministic 1-local routing algorithm is
o(n)-competitive on all pairs of vertices of the constrained
Θ6-graph. After that, we show how to route between any two visible
vertices of the constrained Θ6-graph using only 1-local information.
Our routing algorithm guarantees that the returned path is 2-competitive.
Additionally, we provide a 1-local 18-competitive routing algorithm for visible
vertices in the constrained half-Θ6-graph, a subgraph of the
constrained Θ6-graph that is equivalent to the Delaunay graph where the
empty region is an equilateral triangle. To the best of our knowledge, these
are the first local routing algorithms in the constrained setting with
guarantees on the length of the returned path