Let P be a set of n vertices in the plane and S a set of noncrossing 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 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 using only 1-local information, while guaranteeing that the returned path has length at most 2 times the Euclidean distance between the source and destination. To the best of our knowledge, this is the first local routing algorithm in the constrained setting with guarantees on the path length.