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arXiv · 2106.16130

Diameter estimates for graph associahedra

Jean Cardinal·Lionel Pournin·Mario Valencia-Pabon
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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.

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BibTeXRIS

Jean Cardinal, Lionel Pournin, Mario Valencia-Pabon. 2021-06-30. Diameter estimates for graph associahedra. https://doi.org/10.1007/s00026-022-00598-z

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