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

Abstract

Two new classes of graphs are introduced within the context of the general principle of duality. This duality principle applies to a single graph, planar or nonplanar. The restricted principle generally presented, which applies to a pair of planar graphs, may be considered a subclass of the general principle. As part of the duality principle, it is necessary to define two dual operations on graphs, reduction and contraction of a graph, where reduction is used to define tree graphs and polygon graphs of a given graph, and contraction is used to obtain rose graphs and cut‐set graphs of a graph. The two new classes of graphs are the rose graph and the bouquet graph, the rose graph being defined as a connected graph containing no cut‐sets, and the bouquet graph as a graph whose connected components are rose graphs. Thus the rose graph and the bouquet graph embody concepts dual to those of the tree graph and the forest graph, respectively. Theorems on these new graphs and examples illustrating algorithms are given, where it is shown, for example, that the family of fundamental cut‐sets can be found more easily using a spanning rose graph rather than a spanning tree graph.