Abstract
Graph associahedra are polytopes realizing the nested complex N(G) on connected subgraphs of a graph G.While all known explicit constructions produce polytopes with the same normal fan, the great variety of fan realizationsof classical associahedra and the analogy between finite type cluster complexes and nested complexes incitedus to transpose S. Fomin and A. Zelevinsky's construction of compatibility fans for generalized associahedra (2003)to graph associahedra. Using a compatibility degree, we construct one fan realization of N(G) for each of its facets.Specifying G to paths and cycles, we recover a construction by F. Santos for classical associahedra (2011) and extendF. Chapoton, S. Fomin and A. Zelevinsky's construction (2002) for type B and C generalized associahedra.
Highlights
To cite this version: Thibault Manneville, Vincent Pilaud
We show that our compatibility degree in path and cycle nested complexes matches the compatibility degree in the type A, B and C cluster complexes of S
The problem is that the compatibility degree sometimes equals zero without reflecting any cardinality
Summary
Fix a graph G with vertex set V, let κ(G) be its set of connected components, and set n := |V| − |κ(G)|. The nested complex N (G) of G can be realized as the boundary complex of a convex polytope [FS05, CD06, Zel, Pos, Vol, DFRS15] This complex is an (n − 1)-dimensional simplicial sphere so that any tube of a maximal tubing can be flipped into another tube, described in the following proposition. Dual compatibility vectors) of all tubes of G with respect to any initial maximal tubing T◦ on G support a complete simplicial fan realizing the nested complex N (G) on G. Conjecture 8 All primal and dual compatibility fans of any graph G are normal fans of convex polytopes.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
