  Graph associahedra are natural generalizations of the classical associahedra.
They provide polytopal realizations of the nested complex of a graph $G$,
defined as the simplicial complex whose vertices are the tubes (i.e. connected
induced subgraphs) of $G$ and whose faces are the tubings (i.e. collections of
pairwise nested or non-adjacent tubes) of $G$. The constructions of M. Carr and
S. Devadoss, of A. Postnikov, and of A. Zelevinsky for graph associahedra are
all based on the nested fan which coarsens the normal fan of the permutahedron.
In view of the combinatorial and geometric variety of simplicial fan
realizations of the classical associahedra, it is tempting to search for
alternative fans realizing graphical nested complexes.
  Motivated by the analogy between finite type cluster complexes and graphical
nested complexes, we transpose in this paper S. Fomin and A. Zelevinsky's
construction of compatibility fans from the former to the latter setting. For
this, we define a compatibility degree between two tubes of a graph $G$. Our
main result asserts that the compatibility vectors of all tubes of $G$ with
respect to an arbitrary maximal tubing on $G$ support a complete simplicial fan
realizing the nested complex of $G$. In particular, when the graph $G$ is
reduced to a path, our compatibility degree lies in $\{-1,0,1\}$ and we recover
F. Santos' Catalan many simplicial fan realizations of the associahedron.
