Simplicial tree-decompositions of infinite graphs. III. The uniqueness of prime decompositions

In this last of three papers on simplicial tree-decompositions of graphs we investigate the extent to which prime factors in such decompositions are unique, or depend on the decomposition chosen. A simple example shows that a prime decomposition of a graph may have superfluous factors, the omission of which leaves a set of factors that can be rearranged into another decomposition of the same graph. As our main result we show that this possibility is the only way in which prime decompositions can vary: we prove that all prime decompositions of a countable graph without such superfluous members have the same set of factors. We also obtain a characterization theorem which identifies these factors among similar subgraphs by their position within the graph considered, independently of their role in any decomposition. The material presented in this paper rests on concepts and results developed in [ 4 ]; in particular, the reader will benefit from familiarity with [ 4, Theorem 3.2 ]. The paper does not depend on [ 5 ], but its results are complementary and in that sense related to those obtained in [ 5 ]. Let G be a graph, σ > 0 an ordinal, and let Bλ be an induced subgraph of G for every λ < σ. The family (Bλ)λ<σ is called a simplicial tree-decomposition of G if (S1) G = ⋃