Bounds on the Diameter of Graph Associahedra

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 logn).

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

[2]  Frank Harary,et al.  Graph Theory , 2016 .

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

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

[5]  Daniel M. Kane,et al.  The geometry of binary search trees , 2009, SODA.

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

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

[8]  Jean-Louis Loday.  Realization of the Stasheff polytope , 2002 .

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

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

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

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

[13]  Erik D. Demaine,et al.  Dynamic optimality - almost [competitive online binary search tree] , 2007, 45th Annual IEEE Symposium on Foundations of Computer Science.

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

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

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

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

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

[19]  Gerard J. Chang,et al.  Quasi-threshold Graphs , 1996, Discret. Appl. Math..

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

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

[22]  Jaroslav Nesetril,et al.  Sparsity - Graphs, Structures, and Algorithms , 2012, Algorithms and combinatorics.

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

[24]  Lionel Pournin,et al.  The asymptotic diameter of cyclohedra , 2014, 1410.5259.

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

[26]  Robert E. Wilber.  Lower Bounds for Accessing Binary Search Trees with Rotations , 1989, SIAM J. Comput..

[27]  L. Pournin.  The diameter of associahedra , 2012, 1207.6296.

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

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

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

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