• 1 Central Institute of Economics and Mathematics of the RAS
• 2 Institute for System Analysis of the RAS

Let $G=(V,E)$ be a finite acyclic directed graph. Being motivated by a study of certain aspects of cluster algebras, we are interested in a class of triangulations of the cone of non-negative flows in $G, \mathcal F_+(G)$. To construct a triangulation, we fix a raming at each inner vertex $v$ of $G$, which consists of two linear orders: one on the set of incoming edges, and the other on the set of outgoing edges of $v$. A digraph $G$ endowed with a framing at each inner vertex is called $framed$. Given a framing on $G$, we define a reflexive and symmetric binary relation on the set of extreme rays of $\mathcal F_+ (G)$. We prove that that the complex of cliques formed by this binary relation is a pure simplicial complex, and that the cones spanned by cliques constitute a unimodular simplicial regular fan $Σ (G)$ covering the entire $\mathcal F_+(G)$.