# Planar order on vertex poset

## TL;DR

This paper introduces the concept of planar order on vertex posets of processive planar graphs, establishing a natural correspondence with the planar order on their edge posets, and extends the concept to general finite posets.

## Contribution

It proves the existence of a natural planar order on vertex posets derived from the edge poset's planar order in processive planar graphs.

## Key findings

- Planar order on vertex posets is equivalent to conjugate order.
- A natural planar order on vertex posets can be induced from edge posets.
- The concept extends to any finite poset, not just graphs.

## Abstract

A planar order is a special linear extension of the edge poset (partially ordered set) of a processive plane graph. The definition of a planar order makes sense for any finite poset and is equivalent to the one of a conjugate order. Here it was proved that there is a planar order on the vertex poset of a processive planar graph naturally induced from the planar order of its edge poset.