Marked tubes and the graph multiplihedron

TL;DR

This paper introduces the graph multiplihedron, a convex polytope derived from a graph G with a face poset based on marked subgraphs, generalizing the multiphihedron and connecting to various geometric and algebraic structures.

Contribution

It constructs and realizes the graph multiplihedron, linking it to quilted disks, Riemann surfaces, operads, and known polytopes like Minkowski sums and permutohedra.

Findings

Provides a convex realization of the graph multiplihedron

Establishes connections to quilted disks and operadic structures

Relates specific examples to Minkowski sums and permutohedra

Abstract

Given a graph G, we construct a convex polytope whose face poset is based on marked subgraphs of G. Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of the multiphihedron, but features of this polytope appear in works related to quilted disks, bordered Riemann surfaces, and operadic structures. Certain examples of graph multiplihedra are related to Minkowski sums of simplices and cubes and others to the permutohedron.