Boolean Operations on Graphs.

Abstract

There are several operations on two graphs G1 and G2 which result in a graph G whose set of points is the cartesian product Vx χ V2, where Vk is the point set of Gk. These include the cartesian product (Sabidussi [11]), the composition (Harary [5], Sabidussi [10]), and the tensor prod uct (Weichsel [14], McAndrew [8], Harary and Trauth [7], and Bru aldi [2]). Some of the operations have been independently rediscovered several times. This has led to considerable ambiguity because of the use of different terminology and notation. It is hoped that our systematic nomenclature based on the usual boolean operations becomes standard. These operations are important for constructing new classes of graphs which in turn may be useful for the recognition and decomposition of graphs and for the determination of structural properties of graphs in terms of their constituent subgraphs. The boolean viewpoint introduced here has served to coordinate the definitions of all known operations and to suggest new ones. The alge braic representation of the adjacency matrix of a graph is most con venient in expressing each boolean operation in terms of its constituent graphs G1 and G2. The purposes of this review article are (i) to develop new boolean operations on two graphs, (ii) to relate these to the various existing opera tions, (iii) to investigate some invariant properties of boolean operations, (iv) to demonstrate the way in which boolean operations are related to one another (v) to provide the conditions for the connectedness of graphs obtained by boolean operations, and (vi) to pose some unsolved problems relating to the automorphism group of such a composite graph. Preliminaries. A graph G consists of a finite set V of points and a set X of lines which is a subset of all unordered pairs of points. Our terminology and nota