The uniqueness of prime decompositions
Reinhard Diestel · 1990
Abstract In this chapter we explore the extent to which simplicial decompositions and simplicial tree-decompositions into primes are unique. Clearly, the extreme case of such uniqueness would be that any two prime decompositions of a graph consisted of the same set of factors, arranged in the same order. The latter, however, is almost never true: even in cases of graphs for which the factors in any two prime decompositions are the same, the order of these factors is likely to vary greatly. We shall therefore not consider this order aspect of uniqueness.