A new planarity test based on 3-connectivity

Duality in Networks: Roses, Bouquets, and Cut‐Sets; Trees, Forests and Polygons

“…In some problems one type of graph will be more efficient for use, whereas the other type will be preferred for a different class of problems. In fact, in most problems, as demonstrated in our papers [10,13], it is necessary to use a combination of the two operations and two types of graphs. Furthermore, there are problems, for example, synthesis problems for nonplanar graphs, where there is no choice: it is required that we work with the concepts of the rose and cut-sets rather than.those of the tree and polygons.…”

“…The importance of the graph obtained by what we will call aontraation has not been fully recognized in the literature on the engineering and computer applications of graph theory. In our own papers on the Shannon Switching Game [10], previously mentioned, and on the planarity of graphs [13], we have defined and used the two operations and the resulting graphs on an equal footing. We have also illustrated the point that these can be used in combination to obtain additional graphs.…”

“…Finally, it is pointed out that both of the operations of contraction and reduction may be applied to a graph G to yield new graphs that are reduaed aontraations (or equivalently, aontraated reduations) of G. It can be shown that a reduced contraction of G is independent of the order in which the branches are shorted and opened. In other words, the following identities hold forT c S =E(G): Such reduced contracted graphs are applied in a previously mentioned algorithm for determining planarity of a graph [13].…”

Graph‐realizability of Matroids

“…However, the present graph-realizability algorithm makes it possible to adopt the planarity testing procedure in [3]. The process of constructing Gk-1 from Gk is either to join two vertices of Gk with a new edge or to split a vertex of Gk into two vertices and join them with a new edge.…”

“…In this paper we give the first algorithmic solution to the problem of realizing a general matroid as a graph. The procedure we use is a generalization of a theorem of Bruno et al [3] for determining whether a graph is planar. Their procedure makes use of the wheel graph; our procedure uses the wheel and whirl matroids.…”