Representations of planar graphs

摘要: This paper shows that every 3-connected planar graph G can be represented as a collection of circles, one circle representing each vertex and face, so that, for edge G, the four circles two endpoints neighboring faces meet at point, furthermore vertex-circles cross face-circles right angles. extends result W. Thurston [The Geometry Topology Three Manifolds, unpublished] and, independently, Andreev. From this we deduce corollaries: (1) The partial order formed by taking vertices, edges, bounded ordered inclusion, is order; (2) One represent its dual simultaneously in plane with straight-line edges answers question first asked Tutte [Proc. LMS, 13 (3) (1963), pp. 743–768].