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Diameter Estimates for Graph Associahedra

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 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 \(\Theta (m)\) 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 graphs and of unbalanced complete bipartite graphs.

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Acknowledgements

The authors thank the referees for very helpful comments and suggestions. This work was partially supported by the French-Belgian PHC Project number 42703TD.

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Correspondence to Lionel Pournin.

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Communicated by Kolja Knauer.

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Cardinal, J., Pournin, L. & Valencia-Pabon, M. Diameter Estimates for Graph Associahedra. Ann. Comb. (2022). https://doi.org/10.1007/s00026-022-00598-z

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