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.

Truemper配置是四种类型的图（即theta，轮，棱柱和金字塔），它们在遗传图类的几个分解定理的证明中起着重要作用。在本文中，我们证明了两个结构定理：一个用于不带theta，轮和棱柱的图作为诱导子图，另一个用于不带theta，轮和棱锥的图作为诱导子图。结果是针对这两类的多项式时间识别算法。在本系列的第二部分中，我们将这些结果推广到没有theta和wheel作为诱导子图的图形，在第三和第四部分中，使用获得的结构，我们为这些图形解决了一些优化问题。