Listing spanning trees of outerplanar graphs by pivot-exchanges

TL;DR

This paper presents an efficient method to list all spanning trees of outerplanar graphs with minimal edge exchanges, forming Hamilton paths on the associated 0/1-polytope, and introduces a greedy algorithm for this purpose.

Contribution

It introduces a simple greedy algorithm to list spanning trees of outerplanar graphs with minimal edge exchanges, efficiently generating Hamilton paths on the spanning tree polytope.

Findings

Listings can be generated in O(n log n) time per tree.

Consecutive spanning trees differ by an exchange of two edges sharing an endpoint.

The listings form Hamilton paths on the spanning tree polytope.

Abstract

We prove that the spanning trees of any outerplanar triangulation $G$ can be listed so that any two consecutive spanning trees differ in an exchange of two edges that share an end vertex. For outerplanar graphs $G$ with faces of arbitrary lengths (not necessarily 3) we establish a similar result, with the condition that the two exchanged edges share an end vertex or lie on a common face. These listings of spanning trees are obtained from a simple greedy algorithm that can be implemented efficiently, i.e., in time $\mathcal{O}(n \log n)$ per generated spanning tree, where $n$ is the number of vertices of $G$. Furthermore, the listings correspond to Hamilton paths on the 0/1-polytope that is obtained as the convex hull of the characteristic vectors of all spanning trees of $G$.