The rotation distance of brooms

The associahedron A(G) of a graph G has the property that its vertices can be thought of as the search trees on G and its edges as the rotations between two search trees. If G is a simple path, then A(G) is the usual associahedron and the search trees on G are binary search trees. Computing distances in the graph of A(G), or equivalently, the rotation distance between two binary search trees, is a major open problem. Here, we consider the different case when G is a complete split graph. In that case, A(G) interpolates between the stellohedron and the permutohedron, and all the search trees on G are brooms. We show that the rotation distance between any two such brooms and therefore the distance between any two vertices in the graph of the associahedron of G can be computed in quasi-quadratic time in the number of vertices of G. (c) 2023 Elsevier Ltd. All rights reserved.