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arXiv · 2409.15793

Listing spanning trees of outerplanar graphs by pivot-exchanges

Nastaran Behrooznia·Torsten Mütze
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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 $G$.

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Nastaran Behrooznia, Torsten Mütze. 2024-09-24. Listing spanning trees of outerplanar graphs by pivot-exchanges. https://arxiv.org/abs/2409.15793

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