arXiv subjects
Publications and source records attributed to Caroline Klivans.
We consider a variety of connections between threshold graphs, shifted complexes, and simplicial complexes naturally formed from a graph. These graphical complexes include the independent set, neighborhood, and dominance complexes. We present a number of structural results and relations among them including new characterizations of the class of threshold graphs.
We study the positive Bergman complex B+(M) of an oriented matroid M, which is a certain subcomplex of the Bergman complex B(M) of the underlying unoriented matroid. The positive Bergman complex is defined so that given a linear ideal I with associated oriented matroid M_I, the positive tropical variety associated to I is equal to the fan over B+(M_I). Our main result is that a certain "fine" subdivision of B+(M) is a geometric realization of the order complex of the proper part of the Las Vergnas face lattice of M. It follows that B+(M) is homeomorphic to a sphere. For the oriented matroid of the complete graph K_n, we show that the face poset of the "coarse" subdivision of B+(K_n) is dual to the face poset of the associahedron A_{n-2}, and we give a formula for the number of fine cells within a coarse cell.