Mutations and (Non-)Euclideaness in oriented matroids
Michael Wilhelmi
TL;DR
The paper investigates how mutations and Euclidean behavior stratify oriented matroids into Mandel, Las Vergnas, Euclidean, and realizable classes, proving that all inclusions are proper by constructing explicit examples. It develops a framework around lexicographic extensions and mutation graphs to control mutations and preserve Euclideaness in many constructions, while also showing that Euclideaness can fail under mutation-flips and that LV is not minor-closed. A key result is that Mandel implies Las Vergnas, but LV is strictly larger and not minor-closed, with rank-4 uniform cases yielding strong lower bounds on adjacent mutations and many Mandel examples among non-Euclidean OM. The work also characterizes when Euclideaness is preserved under various operations and identifies several open questions about the structure and closure properties of Mandel and related classes, particularly in higher ranks and under duality.
Abstract
We call an oriented matroid Mandel if it has an extension in general position which makes all programs with that extension Euclidean. If $L$ is the minimum number of mutations adjacent to an element of the groundset, we call an oriented matroid Las Vergnas if $L > 0$. If $\frak{O}_{\mathcal{property}}$ is the class of oriented matroids having a certain property, it holds $\frak{O} \supset \frak{O}_{\mathcal{Las Vergnas}} \supset \frak{O}_{\mathcal{Mandel}} \supset \frak{O}_{\mathcal{Euclidean}} \supset \frak{O}_{\mathcal{realizable}}.$ All these inclusions are proper, we give explicit proofs/examples for the parts of this chain that were not known. For realizable hyperplane arrangements of rank $r$ we have $L = r$ which was proved by Shannon. Under the assumption that a (modified) intersection property holds we give an analogon to Shannons proof and show that uniform rank $4$ Euclidean oriented matroids with that property have $L = 4$. Using the fact that the lexicographic extension creates and destroys certain mutations, we show that for Euclidean oriented matroids holds $L \ge 3$. We give a survey of preservation of Euclideaness and prove that Euclideaness remains after a certain type of mutation-flips. This yields that a path in the mutation graph from a Euclidean oriented matroid to a totally non-Euclidean oriented matroid (which has no Euclidean oriented matroid programs) must have at least three mutation-flips. Finally, a minimal non-Euclidean or rank $4$ uniform oriented matroid is Mandel if it is connected to a Euclidean oriented matroid via one mutation-flip, hence we get many examples for Non-Euclidean but Mandel oriented matroids and have $L \le 3$ for those of rank $4$.