Listing spanning trees of outerplanar graphs by pivot-exchanges
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913620314554368 |
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| author | Behrooznia, Nastaran Mütze, Torsten |
| author_facet | Behrooznia, Nastaran Mütze, Torsten |
| contents | 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 $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Listing spanning trees of outerplanar graphs by pivot-exchanges Behrooznia, Nastaran Mütze, Torsten Discrete Mathematics Combinatorics 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 $G$. |
| title | Listing spanning trees of outerplanar graphs by pivot-exchanges |
| topic | Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2409.15793 |