HOW TO DRAW A GRAPH By W. T. TUTTE [Received 22 May 1962] 1. Introduction WE use the definitions of (11). However, in deference to some recent attempts to unify the terminology of graph theory we replace the term 'circuit' by 'polygon', and 'degree' by 'valency'. A graph G is 3-connected (nodally 3-connected) if it is simple and non-separable and satisfies the following condition; if G is the union of two proper subgraphs H and K such that HnK consists
solely of two vertices u and v, then one of H and K is a link-graph (arc-graph) with ends u and v. It should be noted that the union of two proper subgraphs H and K of G can be the whole of G only if each of H and K includes at least one edge or vertex not belonging to the other. In this paper we are concerned mainly with nodally 3-connected graphs, but a specialization to 3-connected graphs is made in § 12. In § 3 we discuss conditions for a nodally 3-connected graph to be planar, and in § 5 we discuss conditions for the existence of Kuratowski subgraphs of a given graph. In §§ 6-9 we show how to obtain a convex representation of a nodally 3-connected graph, without Kuratowski subgraphs, by solving a set of linear equations. Some extensions of these results to general graphs, with a proof of Kuratowski's theorem, are given in §§ 10-11. In § 12 we discuss the representation in the plane of a pair of dual graphs, and in § 13 we draw attention to some unsolved problems. 2. Peripheral polygons In this section we use the 'nodes' and 'branches'...
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