P-graph Associahedra and Hypercube Graph Associahedra

TL;DR

This paper introduces hypercube-graph associahedra, a new class of polytopes derived from graph tubings with dashed edges, generalizing classical graph associahedra and exploring their combinatorial and geometric properties.

Contribution

It defines hypercube-graph associahedra using tubes and tubings with dashed edges, extending classical graph associahedra, and develops methods to enumerate their face polynomials.

Findings

Characterization of $ riangle$-nested complexes and P-nestohedra.

Properties of hypercube-graph associahedra including facets, faces, and normal fans.

Enumeration methods for $f$-polynomials of hypercube-graph associahedra.

Abstract

A graph associahedron is a polytope dual to a simplicial complex whose elements are induced connected subgraphs called tubes. Graph associahedra generalize permutahedra, associahedra, and cyclohedra, and therefore are of great interest to those who study Coxeter combinatorics. This thesis characterizes nested complexes of simplicial complexes, which we call $\Delta$-nested complexes. From here, we can define P-nestohedra by truncating simple polyhedra, and in more specificity define P-graph associahedra, which are realized by repeated truncation of faces of simple polyhedra in accordance with tubes of graphs. We then define hypercube-graph associahedra as a special case. Hypercube-graph associahedra are defined by tubes and tubings on a graph with a matching of dashed edges, with tubes and tubings avoiding those dashed edges. These simple rules make hypercube-graph tubings a simple and intuitive extension of classical graph tubings. We explore properties of $\Delta$-nested complexes and P-nestohedra, and use these results to explore properties of hypercube-graph associahedra, including their facets and faces, as well as their normal fans and Minkowski sum decompositions. We use these properties to develop general methods of enumerating $f$-polynomials of families of hypercube-graph associahedra. Several of these hypercube-graphs correspond to previously-studied polyhedra, such as cubeahedra, the halohedron, the type $A_n$ linear $c$-cluster associahedron, and the type $A_n$ linear $c$-cluster biassociahedron. We provide enumerations for these polyhedra and others.