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.
翻译:osociaedra 的图形是通用的 motoohedra, 以巢穴和高地的顶点为特例。 一个图形的图形 : $G$ 以G$编码搜索树的组合体, 由根 $- r$ 重新定义。 特别是, 图形的骨架是这些搜索树的旋转图。 我们根据某些图形参数的函数来调查图 : osoahedra 的直径。 我们给小的完美图形直径以$(m) $G$ 以G$为单位, 由根 $- 美元和 $- r$ 连接的每个组件的搜索树状图, 以及搜索树深度、 树width 或路径widt 的树骨架的最大直径。 我们还证明, 路径光线直径图的完整ncial- naddra 和正平面的正平面图的直径是 。