# Finding Tutte paths in linear time

## TL;DR

This paper presents a new, simpler proof and a linear-time algorithm for finding Tutte paths in 3-connected planar graphs, with applications to spanning trees and walks.

## Contribution

It provides a new proof and a linear-time algorithm for Tutte paths in 3-connected planar graphs, with enhanced properties and applications.

## Key findings

- Linear-time algorithm for Tutte paths in 3-connected planar graphs
- Tutte paths visit all exterior vertices and have exactly three attachment points per component
- Applications include binary spanning trees and 2-walks

## Abstract

It is well-known that every planar graph has a Tutte path, i.e., a path $P$ such that any component of $G-P$ has at most three attachment points on $P$. However, it was only recently shown that such Tutte paths can be found in polynomial time. In this paper, we give a new proof that 3-connected planar graphs have Tutte paths, which leads to a linear-time algorithm to find Tutte paths. Furthermore, our Tutte path has special properties: it visits all exterior vertices, all components of $G-P$ have exactly three attachment points, and we can assign distinct representatives to them that are interior vertices. Finally, our running time bound is slightly stronger; we can bound it in terms of the degrees of the faces that are incident to $P$. This allows us to find some applications of Tutte paths (such as binary spanning trees and 2-walks) in linear time as well.