Simplicial tree-decompositions of infinite graphs. III: The uniqueness of prime decompositions (Q750465)

A simplicial tree-decomposition of a graph is a covering family of induced subgraphs (factors), indexed by some ordinal, which overlap only by simplices (complete subgraphs). A prime decomposition is one where each factor is prime, i.e. not further decomposable. In this third paper on the subject the unicity of prime decompositions is studied. It turns out that although the order of the factors is not necessarily unique, the attaching simplices are: these are the complete subgraphs which minimally separate two vertices. As to the factors in a prime decomposition, only some are indispensable, in the sense that they may not be removed. There are two reasons only for indispensability: necessity to obtain a covering, and necessity for connectivity. A prime decomposition is reduced when only indispensable factors are present. If a countable graph admits a reduced prime-decomposition, its factors are uniquely determined.