Encompassing colored planar straight line graphs

Consider a planar straight line graph (PSLG), G, with k connected components, k⩾2. We show that if no component is a singleton, we can always find a vertex in one component that sees an entire edge in another component. This implies that when the vertices of G are colored, so that adjacent vertices have different colors, then (1) we can augment G with k−1 edges so that we get a color conforming connected PSLG; (2) if each component of G is 2-edge connected, then we can augment G with 2k−2 edges so that we get a 2-edge connected PSLG. Furthermore, we can determine a set of augmenting edges in O(nlogn) time. An important special case of this result is that any red–blue planar matching can be completed into a crossing-free red–blue spanning tree in O(nlogn) time.