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Compatibility fans for graphical nested complexes. (English) Zbl 1362.05140
Summary: Graph associahedra generalize classical associahedra. They realize the nested complex of a graph $$G$$, i.e. the simplicial complex whose vertices are the tubes (connected induced subgraphs) of $$G$$ and whose faces are the tubings (collections of pairwise nested or non-adjacent tubes) of $$G$$. The constructions of M. Carr and S. L. Devadoss [Topology Appl. 153, No. 12, 2155–2168 (2006; Zbl 1099.52001)], of A. Postnikov [Int. Math. Res. Not. 2009, No. 6, 1026–1106 (2009; Zbl 1162.52007)], and of A. Zelevinsky [Pure Appl. Math. Q. 2, No. 3, 655–671 (2006; Zbl 1109.52010)] for graph associahedra are all based on the nested fan, which coarsens the normal fan of the permutahedron. In view of the variety of fan realizations of associahedra, it is tempting to look for alternative fans realizing graphical nested complexes. Motivated by the analogy between finite type cluster complexes and graphical nested complexes, we transpose S. Fomin and A. Zelevinsky’s [Invent. Math. 154, No. 1, 63–121 (2003; Zbl 1054.17024)] compatibility fans from the former to the latter setting. We define a compatibility degree between two tubes of a graph $$G$$ and show that the compatibility vectors of all tubes of $$G$$ with respect to an arbitrary maximal tubing on $$G$$ support a fan realizing the nested complex of $$G$$. When $$G$$ is a path, we recover F. Santos’ Catalan many realizations of the associahedron.

##### MSC:
 05E45 Combinatorial aspects of simplicial complexes 05B45 Combinatorial aspects of tessellation and tiling problems 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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