Let S be a finite set of points in the interior of a simple polygon P. A geodesic graph, $G_P(S,E)$, is a graph with vertex set S and edge set E such that each edge $(a,b)\in E$ is the shortest path between a and b inside P. $G_P$ is said to be plane if the edges in E do not cross. If the points in S are colored, then $G_P$ is said to be properly colored provided that, for each edge $(a,b)\in E$, a and b have different colors. In this paper we consider the problem of computing (properly colored) plane geodesic perfect matchings, Hamiltonian cycles, and spanning trees of maximum degree three.