Problem of the Week
Hilton Memorial Lecture
The acyclotope of a graph G can be defined as the convex hull of the net degree vectors (indegree − outdegree) of all acyclic orientations of G, or as the Minkowski sum of line segments representing G. (Thus, it is a zonotope; Postnikov calls it the “graphical zonotope”. The line segments are dual to the hyperplanes of the graphic arrangement of G.) I introduced the acyclotope (or at least the name) about 37 1/2 years ago in connection with signed graphs. It was perhaps inspired by the score vectors of tournaments, which are a special case that is closely related to the permutahedron. I will discuss its notable properties.