Root polytopes and Jaeger‐type dissections for directed graphs

We associate root polytopes to directed graphs and study them by using ribbon structures. Most attention is paid what we call the semi-balanced case, that is, when each cycle has same number of edges pointing in two directions. Given a structure, identify natural class spanning trees show that, they induce shellable dissection polytope into maximal simplices. This allows for computation h ∗ $h^*$ -vector showing some properties this new graph invariant, such as product formula planar equivalent greedoid polynomial dual graph. obtain general recursion relation well. also work out case layer-complete graphs, where our method recovers previously known triangulation. Indeed often but not always triangulation; address with series examples.