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Bounds on the Diameter of Graph Associahedra

par Cardinal, Jean ;Pournin, Lionel;Valencia-Pabon, Mario M.
Référence Procedia Computer Science, 195, page (239-247)
Publication Publié, 2021

Article révisé par les pairs

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Résumé :

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. It is known that the diameter of the associahedra of paths of length n, the classical associahedra, is 2n - 6 for a large enough n. 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. Finally, we prove that the maximum diameter of associahedra of graphs of pathwidth two is Θ(n log n).