A short proof of Kuratowski's graph planarity criterion

We present a new short combinatorial proof of the sufficiency part of the well-known Kuratowski's graph planarity criterion. The main steps are to prove that for a minor minimal non-planar graph G and any edge xy: (1) G-x-y does not contain θ-subgraph; (2) G-x-y is homeomorphic to the circle; (3) G is either K5 or K{3,3}. c © 1997 John Wiley & Sons, Inc. In 1930, K. Kuratowski published his well-known graph planarity criterion [1]: a graph is planar if and only if it does not contain a subgraph, homeomorphic to eitherK5 orK{3,3}. Since then, many new and shorter proofs of this criterion appeared [2]. In this paper we present a short combinatorial proof of the ‘‘if’’part. It is based on contracting edge, similar to that of [2, section 5], but we avoid the reduction to 3-connected graphs. By θ-subgraph we mean a subgraph homeomorphic to K{3,2}. Consider a minor minimal non-planar graph G. Lemma 1. If xy E(G), then G-x-y does not contain a θ-subgraph. Proof. Suppose not. Consider an embedding of G/xy in the plane. Let G′ = G-x-y = (G/xy)-(xy). Let F be the subgraph ofG′ bounding the face ofG′ containing the deleted vertex xy of G/xy. Then F cannot contain a θ-subgraph [2, section 1]. But since G′ does, there is an edge e inE(G′)−E(F ). Since for each forest T ⊆ R, R −T is connected, F contains a cycle Journal of Graph Theory Vol. 25, 129 131 (1997) c © 1997 John Wiley & Sons, Inc. CCC 0364-9024/97/020129-03 130 JOURNAL OF GRAPH THEORY