Diameter Estimates for Graph Associahedra

Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph G encodes the combinatorics of the search trees on G , defined recursively by a root r together with search trees on each of the connected components of G-r . In particular, the 1-skeleton of the corresponding graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. We give a tight bound of Θ (m) on the diameter of trivially perfect graph associahedra on m edges. We consider the maximum diameter of associahedra of graphs on n vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. We also prove that the maximum diameter of associahedra of graphs of pathwidth two is Θ (nlog n) . Finally, we give the exact diameter of the associahedra of complete split graphs and of unbalanced complete bipartite graphs.