A sufficient condition for the existence of plane spanning trees on geometric graphs ?
Let P  be a set of n?3 n ? 3 points in general position in the plane and let G be a geometric graph with vertex set P . If the number of empty triangles ?uvw ? u v w in P for which the subgraph of G  induced by {u,v,w} { u , v , w } is not connected is at most n?3 n ? 3 , then G contains a non-self intersecting spanning tree.
geometric graph empty triangle plane tree