Markov bases of decomposable models for contingency tables

We study Markov bases of hierarchical models in general and those of decomposable models in particular for multiway contingency tables by determining the structure of fibers of sample size two. We prove that the number of elements of fibers of sample size two are powers of two and we characterize the primitive moves (i.e. square-free moves of degree two) of Markov bases in terms of connected components of the independence graph of the generating class of a hierarchical model. This allows us to derive a complete description of minimal Markov bases and minimal invariant Markov bases for decomposable models in view of the fact that they posses Markov bases consisting of primitive moves.