On Tope Graphs of Complexes of Oriented Matroids

Abstract

We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third characterization in terms of zone graphs of tope graphs.Further corollaries include a characterization of topes of oriented matroids due to da Silva, another one of Handa, a characterization of lopsided systems due to Lawrence, and an intrinsic characterization of tope graphs of affine oriented matroids. Moreover, we obtain purely graph theoretic polynomial time recognition algorithms for tope graphs of the above and a finite list of excluded partial cube minors for the bounded rank case.In particular, our results answer a relatively long-standing open question in oriented matroids and can be seen as identifying the theory of (complexes of) oriented matroids as a part of metric graph theory. Another consequence is that all finite Pasch graphs are tope graphs of complexes of oriented matroids, which confirms a conjecture of Chepoi and the two authors.

Two-Dimensional Partial Cubes

“…3 ) [18]), median graphs (F(Q − 3 , C 6 ) [18]), bipartite cellular graphs (F(Q − 3 , Q 3 ) [18]), rank two COMs (F(SK 4 , Q 3 ) [34]), and two-dimensional ample graphs (F(C 6 , Q 3 ) [34]). Here Q − 3 denotes the 3-cube Q 3 with one vertex removed and SK 4 the full subdivision of K 4 , see Figure 3.…”

“…For example, 1-inclusion graphs of ample families (and thus of maximum families) are partial cubes [5,35] (in view of this, we will call such graphs ample partial cubes and maximum partial cubes, respectively). Other important examples comprise median graphs (aka 1-skeletons of CAT(0) cube complexes [17,48]) and, more generally, 1-skeletons of CAT(0) Coxeter zonotopal complexes [28], the tope graphs of oriented matroids (OMs) [8], of affine oriented matroids (AOMs) [34], and of lopsided sets (LOPs) [34,35], where the latter coincide with ample partial cubes (AMPs). More generally, tope graphs of complexes of oriented matroids (COMs) [6,34] capture all of the above.…”

“…Other important examples comprise median graphs (aka 1-skeletons of CAT(0) cube complexes [17,48]) and, more generally, 1-skeletons of CAT(0) Coxeter zonotopal complexes [28], the tope graphs of oriented matroids (OMs) [8], of affine oriented matroids (AOMs) [34], and of lopsided sets (LOPs) [34,35], where the latter coincide with ample partial cubes (AMPs). More generally, tope graphs of complexes of oriented matroids (COMs) [6,34] capture all of the above. Other classes of graphs defined by distance or convexity properties turn out to be partial cubes: bipartite cellular graphs (aka bipartite graphs with totally decomposable metrics) [3], bipartite Pasch [14,16] and bipartite Peano [46] graphs, netlike graphs [45], and hypercellular graphs [18].…”

On antipodal and diametrical partial cubes

“…They were later studied by Parthasarathy and Nandakumar [17] under the name of self-centered unique eccentric point graphs, then by Göbel and Veldman [8] under the name of even graphs, by Fukuda and Handa [6] who proved that the tope graphs of oriented matroids are harmonic partial cubes (i.e., isometric subgraphs of hypercubes), and finally by Klavžar and Kovše [11] who gave a partial solution to a problem set in [6]. Partial cubes, which were introduced by Firsov [5] and characterized by Djoković [4] and Winkler [20], have been extensively studied, see [11,15,12,18] for recent papers. In [11,12,18], antipodal, diametrical and harmonic partial cubes play a very important role.…”

“…Partial cubes, which were introduced by Firsov [5] and characterized by Djoković [4] and Winkler [20], have been extensively studied, see [11,15,12,18] for recent papers. In [11,12,18], antipodal, diametrical and harmonic partial cubes play a very important role.…”