  Truemper configurations are four types of graphs (namely thetas, wheels,
prisms and pyramids) that play an important role in the proof of several
decomposition theorems for hereditary graph classes. In this paper, we prove
two structure theorems: one for graphs with no thetas, wheels and prisms as
induced subgraphs, and one for graphs with no thetas, wheels and pyramids as
induced subgraphs. A consequence is a polynomial time recognition algorithms
for these two classes. In Part II of this series we generalize these results to
graphs with no thetas and wheels as induced subgraphs, and in Parts III and IV,
using the obtained structure, we solve several optimization problems for these
graphs.
