Characterization of single-element extensions preserving Euclideanness

Characterize which kinds of single-element extensions of oriented matroids preserve the Euclidean property of an oriented matroid program.

Background

Euclidean oriented matroid programs are defined by the absence of directed cycles in their directed cocircuit graphs. Although Euclideanness is known to be preserved under taking minors and parallel extensions, its behavior under general single-element extensions was not understood. The paper addresses an important subclass by proving that lexicographic extensions preserve Euclideanness, but the broader classification of extensions that preserve this property remains unresolved.