On generating planar graphs

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Abstract

A 3-valent graph G is cyclically n-connected provided one must cut at least n edges in order to separate any two circuits of G. If G is cyclically n-connected but any separation of G by cutting n edges yields a component consisting of a simple circuit, then we say that G is strongly cyclically n-connected. We prove that there exists a graph G0 such that all strongly cyclically 4-connected planar graphs, other than the graph of the cube and the pentagonal prism, can be generated from G0 by adding edges. We introduce two operations called adding a pair of edges and replacing a face. We prove that using these two operations together with adding edges we can generate the cyclically 5-connected planar graphs.

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Research supported by N.S.F. Grant GP-8470.