Diameter estimates for graph associahedra

Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. ‘e graph associahedron of a graph G encodes the combinatorics of search trees on G, de€ned 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 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. We also prove that the maximum diameter of associahedra of graphs of pathwidth two is Θ(n logn). Finally, we give the exact diameter of the associahedra of complete split and of unbalanced complete bipartite graphs.

[1]  Robert E. Tarjan,et al.  Self-adjusting binary search trees , 1985, JACM.

[2]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[3]  Jean Cardinal,et al.  Competitive Online Search Trees on Trees , 2019, SODA.

[4]  M. Farber,et al.  Convexity in graphs and hypergraphs , 1986 .

[5]  Robert E. Tarjan,et al.  Rotation distance, triangulations, and hyperbolic geometry , 1986, STOC '86.

[6]  E. S. Wolk.  The comparability graph of a tree , 1962 .

[7]  Vincent Pilaud,et al.  GRAPH PROPERTIES OF GRAPH ASSOCIAHEDRA , 2014, 1409.8114.

[8]  N. Erokhovets.  Gal’s conjecture for nestohedra corresponding to complete bipartite graphs , 2009 .

[9]  Benjamin Aram Berendsohn,et al.  Splay trees on trees , 2022, Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA).

[10]  Michal Pilipczuk,et al.  Exploring the Subexponential Complexity of Completion Problems , 2015, TOCT.

[11]  Klaus Jansen,et al.  Rankings of Graphs , 1998, SIAM J. Discret. Math..

[12]  Satyan L. Devadoss,et al.  Coxeter Complexes and Graph-Associahedra , 2004, math/0407229.

[13]  Jim Stasheff,et al.  Homotopy associativity of $H$-spaces. II , 1963 .

[14]  Robert E. Wilber.  Lower bounds for accessing binary search trees with rotations , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[15]  Alan H. Karp.  Bit Reversal on Uniprocessors , 1996, SIAM Rev..

[16]  A. Postnikov,et al.  Faces of Generalized Permutohedra , 2006, math/0609184.

[17]  Satyan L. Devadoss.  A realization of graph associahedra , 2006, Discret. Math..

[18]  Martin Charles Golumbic,et al.  Trivially perfect graphs , 1978, Discret. Math..

[19]  Suzanne M. Seager,et al.  Ordered colourings , 1995, Discret. Math..

[20]  A. Lauve,et al.  New Hopf Structures on Binary Trees , 2009 .

[21]  R. E. Bixby,et al.  The Partial Order of a Polymatroid Extreme Point , 1985, Math. Oper. Res..

[22]  Alejandro A. Schäffer,et al.  Optimal Node Ranking of Trees in Linear Time , 1989, Inf. Process. Lett..

[23]  Erik D. Demaine,et al.  Dynamic Optimality - Almost , 2004, FOCS.

[24]  Francisco Santos,et al.  A counterexample to the Hirsch conjecture , 2010, ArXiv.

[25]  T. Januszkiewicz,et al.  Fundamental Groups of Blow-ups , 2002 .

[26]  Jaroslav Nesetril,et al.  On Low Tree-Depth Decompositions , 2014, Graphs Comb..

[27]  R. Tarjan,et al.  Rotation distance, triangulations, and hyperbolic geometry , 1986, STOC '86.

[28]  Alexander Postnikov,et al.  Permutohedra, Associahedra, and Beyond , 2005, math/0507163.

[29]  Jean Cardinal,et al.  On the Diameter of Tree Associahedra , 2018, Electron. J. Comb..