Non-planar Extensions of Planar Graphs

A graph G is almost 4-connected if it is simple, 3-connected, has at least ve vertices, and V (G) cannot be partitioned into three sets A;B;C in such a way that jCj = 3, jAj 2, jBj 2, and no edge of G has one end in A and the other end in B. A graph K is a subdivision of a graph G if K is obtained from G by replacing its edges by internally disjoint nonzero length paths with the same ends, called segments. Let G;H be almost 4connected graphs such that G is planar, H is non-planar, and H has a subgraph isomorphic to a subdivision of G. If K is a subgraph of H, then a K-path in H is a path with at least one edge, both ends in K, and otherwise disjoint from K. We prove that there exists a subgraph K of H isomorphic to a subdivision of G such that one of the following conditions holds. (i) There exists a K-path in H such that no region boundary of K contains both ends of the path. (ii) There exist two disjoint K-paths with ends s1; t1 and s2; t2, respectively, such that the vertices s1; s2; t1; t2 belong to some region boundary ofK in the order listed. Moreover, for i = 1; 2 the vertices si and ti do not belong to the same segment of K, and if two segments of K include all of s1; t1; s2; t2, then those segments are vertex-disjoint. This is a lemma to be used in later papers. 2