Complete enumeration of small realizable oriented matroids

Point configurations and convex polytopes play central roles in computational geometry and discrete geometry. For many problems, their combinatorial structures, i.e., the underlying oriented matroids up to isomorphism, are often more important than their metric structures. For example, the convexity, the face lattice of the convex hull and all possible triangulations of a given point configuration are determined by its combinatorial structure. One of the most significant merits to consider combinatorial types of them is that there are a finite number of them for any fixed sizes (dimension and number of elements) while there are infinitely many those objects. This enables us to enumerate those objects and study them through computational experiments (for example, see [1, 2, 12]). Despite its merits, enumerating combinatorial types of point configurations is known to be a quite hard task. In fact, they do not admit good combinatorial characterizations unless P = NP [20]. On the other hand, this problem can be overcome in an abstract combinatorial setting of oriented matroids, denoted by OMs shortly. In fact, Finschi and Fukuda [12] performed a large-scale enumeration of OMs including non-uniform ones and of high rank. Aichholzer, Aurenhammer and Krasser [1], and Aichholzer and Krasser [2] enumerated a large class of rank-3 uniform OMs, non-degenerate configurations in the abstract setting. Now, to obtain all possible combinatorial types of point configurations from OM catalogues OM(r, n), the set of all OMs of rank r on n elements, we only need to extract those OMs that are acyclic and realizable. Formally, the realizability problem is to decide whether a given OM can be realized by a vector configuration or not, which is the most crucial part to detect the combinatorial types of point configurations. It is as difficult