Persistent Graphs and Cyclic Polytope Triangulations

TL;DR

This paper establishes a bijection between triangulations of 3D cyclic polytopes and persistent graphs, linking combinatorial structures and providing an efficient enumeration algorithm.

Contribution

It introduces a novel bijection connecting polytope triangulations with persistent graphs and relates these to higher Bruhat orders, along with an enumeration algorithm.

Findings

Bijection between triangulations and persistent graphs

Connection to the second higher Bruhat order

Efficient enumeration algorithm for persistent graphs

Abstract

We prove a bijection between the triangulations of the 3-dimensional cyclic polytope C(n+2, 3) and persistent graphs with n vertices. We show that under this bijection the Stasheff-Tamari orders on triangulations naturally translate to subgraph inclusion between persistent graphs. Moreover, we describe a connection to the second higher Bruhat order B(n, 2). We additionally give an algorithm to efficiently enumerate all persistent graphs on n vertices and thus all triangulations of C(n+2, 3).