A graph-based algorithm for generating the Delaunay triangulation of aplanar point set is presented in this paper. The first step is to calculate the Euclidean Minimum Spanning Tree (EM ST) of the given points. The EM ST is then augumented to triangle mesh. Finally the Delaunay Triangulation of the given points is obtained by local tranformation acording to Max-M in angle criteria. The concepts and properties of Vorono i diagram of a planar point set are slso discussed in the paper.