Diameter estimates for graph associahedra

TL;DR

This paper studies the diameter of graph associahedra, providing tight bounds for specific graph classes and parameters, and exact diameters for certain special graphs, advancing understanding of their combinatorial complexity.

Contribution

It offers new bounds and exact values for the diameter of graph associahedra based on various graph parameters, including for trivially perfect graphs and graphs with pathwidth two.

Findings

Diameter of trivially perfect graph associahedra is Θ(m).

Maximum diameter for graphs with pathwidth two is Θ(n log n).

Exact diameters computed for complete split and unbalanced bipartite graphs.

Abstract

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 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 skeleton of the 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 $\Theta(m)$ 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 $\Theta (n\log n)$. Finally, we give the exact diameter of the associahedra of complete split and of unbalanced complete bipartite graphs.