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We associate root polytopes to directed graphs and study them by using ribbon structures. Most attention is paid to what we call the semi-balanced case, i.e., when each cycle has the same number of edges pointing in the two directions. Given a ribbon structure, we identify a natural class of spanning trees and show that, in the semi-balanced case, they induce a shellable dissection of the root polytope into maximal simplices. This allows for a computation of the $h^*$-vector of the polytope and for showing some properties of this new graph invariant, such as a product formula and that in the planar case, the $h^*$-vector is equivalent to the greedoid polynomial of the dual graph. We obtain a general recursion relation as well. We also work out the case of layer-complete directed graphs, where our method recovers a previously known triangulation. Indeed our dissection is often but not always a triangulation; we address this with a series of examples.
Bibtex entry:
| AUTHOR | = | {K{\'a}lm{\'a}n, Tam{\'a}s and T{\'o}thm{\'e}r{\'e}sz, Lilla}, |
| TITLE | = | {Root polytopes and Jaeger-type dissections for directed graphs}, |
| NOTE | = | {{\tt egres.elte.hu}}, |
| INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
| YEAR | = | {2022}, |
| NUMBER | = | {TR-2022-16} |