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.
A wheel is a graph made of a cycle of length at least 4 together with a vertex that has at least three neighbors in the cycle. We prove that the problem whose instance is a graph G and whose question is "does G contains a wheel as an induced subgraph" is NP-complete. We also settle the complexity of several similar problems.
In this paper we investigate the structural properties of kpath separable graphs, that are the graphs that can be separated by a set of k shortest paths. We identify several graph families having such path separability, and we show that this property is closed under minor taking. In particular we establish a list of forbidden minors for 1-path separable graphs. The second author is also member of the "Institut Universitaire de France". Both authors are supported by the ANR-project "ALADDIN", and the équipe-projet commune LaBRI-INRIA Bordeaux Sud-Ouest "CÉPAGE".
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