Algebraic structures on graph associahedra

M. Carr and S. Devadoss introduced in [7] the notion of tubing on a finite simple graph $\Gamma$, in the context of configuration spaces on the Hilbert plane. To any finite simple graph $\Gamma$ they associated a finite partially ordered set, whose elements are the tubings of $\Gamma$ and whose geometric realization is a convex polytope ${\mathcal K}\Gamma$, the graph-associahedron. For the complete graphs they recovered permutahedra, for linear graphs they got Stasheff's associahedra, while for simple graph they obtained the standard simplexes. The goal of the present work is to give an \emph{algebraic} description of graph associahedra. We introduce a substitution operation on tubings, which allows us to describe the set of faces of graph-associahedra as a free object, spanned by the set of all connected simple graphs, under operations given via connected subgraphs. The boundary maps of graph-associahedra defines natural derivations in this context. Along the way, we introduce a topological interpretation of the graph tubings and our new operations. In the last section, we show that substitution of tubings may be understood in the context of M. Batanin and M. Markl's operadic categories.