A sufficient condition for the existence of plane spanning trees on geometric graphs

Let P   be a set of n⩾3n⩾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 ΔuvwΔuvw in P for which the subgraph of G   induced by {u,v,w}{u,v,w} is not connected is at most n−3n−3, then G contains a non-self intersecting spanning tree.