Algebraic structures on graph associahedra

TL;DR

This paper provides an algebraic framework for understanding graph associahedra, introducing operations on tubings that describe their face structures and connect to operadic categories, extending previous geometric and combinatorial insights.

Contribution

It introduces a substitution operation on tubings, enabling an algebraic description of graph associahedra as free objects generated by connected graphs.

Findings

Algebraic description of graph associahedra via substitution operations

Faces of graph-associahedra form a free object under these operations

Connection of substitution operations to operadic categories

Abstract

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.