Ordinals and graph decompositions

A result of Diestel says that every countable simplicial tree decomposition can be rearranged to have length at most ω. We show that no such ordinal bound can be found for the lengths of non-tree decompositions. More generally, we show that for each ordinal σ, there is a decomposable graph whose shortest simplicial decomposition has length exactly σ. Adapting this argument, we show that the index set of decomposable computable graphs DECOMP is Π11 hard by showing that WO is 1-reducible to DECOMP.