平面图与对偶原理

该文利用对偶原理创造性地解决了平面图、连通图及对偶图之间的相互关系问题,纠正了长期以来对于平面图及其同构的错误认识,指出平面图必为连通图,平面图本质上是画在同一平面上的顶点、边、面均不相交的连通图。两个平面图的同构指这两个平面图的顶点、边、面之间均有一一对应关系。面是平面图区别于非平面图的本质特征。同构的平面图的对偶图必同构,事实上,平面图的对偶图是唯一的。任意一个平面图都伴有一个隐图,而该隐图实质上是该平面图的对偶图,该隐图可(根据对偶原理)通过D—过程画出。平面图与其对偶图互为对偶。显平面图与其隐对偶图合称为相伴对偶图。

Planar Graph and the Principle of Duality

Using the principle of duality,the article creatively solves the problem of relationships between planar graphs,connected graphs and dual graphs.It points out and corrects the long-standing mistakes about planar graphs and their isomorphism.A planar graph is essentially a connected graph,in fact,it is a connected graph drawn on a plane with its vertices,edges and faces having no intersection.The isomorphism of two planar graphs refers to the relationship that they have one-to-one correspondence between their vertices,edges and faces.Face is an essential characteristic that distinguishes planar graphs from nonplanar graphs.The dual graphs of isomorphic graphs are also essentially isomorphic;in fact,the dual graph of a planar graph is unique.A planar graph has a corresponding invisible graph,in fact,this invisible graph is its dual graph.It can be drawn out through D-process(based on the principle of duality).A planar graph and its dual graph are mutually dual graphs.A visible planar graph and its invisible dual graph are collectively called mating dual graphs.