Application of Cotree in Topological Classification of Graphlike Manifold
Qing Chen · Keji daobao · 2009
The graphlike manifold M(G) is a tubular surface with the frame of a simple connected graph G. This paper studies the enumeration problem of homeomorphic equivalence classes of the graphlike manifold, which can be transformed into a 2-edge-coloring enumeration problem for graphs. A new approach is proposed in this paper, based on the cut space and the cycle space in the graph theory. It is well known that the enumeration theory of graphs is composed of two parts: the enumeration of labeled graphs and that of unlabelled graphs. Generally speaking, the former is easier than the latter (as the latter involves isomorphism transformation). The enumeration formula and related results of labeled graphs are obtained by means of the group theory in the paper. The enumeration based on the cycle space and the cut space in the graph theory not only simplifies the proofs based on the group theory, but also better reveals the relation between homeomorphic classes and the cycle-structure of graphs. Based on the definition of a new kind, named cotree scheme, is proposed. It is found that a transverse of T-classes can be constructed by taking all edge-induced subgraphs of the as the black subgraphs. A representative system of H-classes can be constructed by taking all non-isomorphic edge-induced subgraphs of the as the black subgraphs.