Combinatorics on Adjacency Graphs and Incidence Pseudographs

Abstract The paper starts with a brief review of combinatorial results for adjacency and oriented adjacency graphs (combinatorial maps). The main subject is incidence pseudographs (a dual of cell complexes) of the n -dimensional orthogonal grid. It is well known that these pseudographs (or complexes) allow a definition of a topological space, and combinatorial formulas are provided for characterizing open and closed sets in this topology. The paper extends work by K. Voss in 1993, which is (in the terminology of incidence pseudographs) on open regions only. The paper also provides combinatorial formulas for closed regions. Matching formulas and Euler characteristic calculations are generalized for arbitrary open or closed regions in the n -dimensional orthogonal grid. This article presents material what will be part of a forthcoming book [11]. The author thanks Azriel Rosenfeld and Klaus Voss for comments which have been important in finalizing this article. The author dedicates this paper to Klaus Voss on the occasion of his retirement in 2002. His outstanding creativity has been very inspiring for all who had or have the opportunity to work with him.