Polytopes of Cographic Unimodular Systems and Valuated Delta-Matroids

Abstract

Here we will cover topics arising from matroid theory. To any graphic matroid, we can associate a certain class of polytopes via its representation forming the well-studied generalized permutahedra. When studying these permutehdra, it has been useful understanding both their normal fans and the support functions. We aim to address this in the more unknown case of the cographic matroids, duals of the graphic matroid. To the associated polytopes, will introduce a characterization for both the normal fans and the support functions. Additionally, with valuated delta matroids, we introduce an object that simultaneously extends valuated matroids and Delta matroids. Both these valuated and delta matroids are themselves a generalization of matroids. We will show that our valuated delta matroids satisfy properties we would expect a generalization should. Furthermore, we discuss the representability of these structures and how this relates to what is already known.