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arXiv · 2105.00960

Root polytopes and Jaeger-type dissections for directed graphs

Tamás Kálmán·Lilla Tóthmérész
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Abstract

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.

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Tamás Kálmán, Lilla Tóthmérész. 2021-05-03. Root polytopes and Jaeger-type dissections for directed graphs. https://doi.org/10.1112/mtk.12163

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