# Generalized cut and metric polytopes of graphs and simplicial complexes

## TL;DR

This paper explores generalized cut and metric polytopes for graphs and simplicial complexes, extending classical concepts and computing these polytopes for various graphs, including oriented and higher-dimensional cases.

## Contribution

It introduces new generalized polytopes for graphs and simplicial complexes, including oriented and higher-dimensional versions, and computes these for many graphs.

## Key findings

- Computed cut and metric polytopes for numerous graphs.
- Defined new oriented and higher-dimensional polytopes.
- Extended classical graph polytopes to simplicial complexes.

## Abstract

Given a graph $G$ one can define the cut polytope CUTP(G) and the metric polytope METP(G) of this graph and those polytopes encode in a nice way the metric on the graph. According to Seymour's theorem, CUTP(G) = METP(G) if and only if K_5 is not a minor of G. We consider possibly extensions of this framework: a) We compute the CUTP(G) and METP(G) for many graphs. b) We define the oriented cut polytope WOMCUTP(G) and oriented multicut polytope OMCUTP(G) as well as their oriented metric version QMETP(G) and WQMETP(G). c) We define an $n$-dimensional generalization of metric on simplicial complexes.