Generating the Spanning Trees of Series-Parallel Graphs up to Graph Automorphism
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908494447247360 |
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| author | Karamchedu, Mithra Bang, Lucas |
| author_facet | Karamchedu, Mithra Bang, Lucas |
| contents | In this paper, we investigate the problem of generating the spanning trees of a graph $G$ up to the automorphisms or "symmetries" of $G$. After introducing and surveying this problem for general input graphs, we present algorithms that fully solve the case of series-parallel graphs, under two standard definitions. We first show how to generate the nonequivalent spanning trees of a oriented series-parallel graph $G$ in output-linear time, where both terminals of $G$ have been individually distinguished (i.e. applying an automorphism that exchanges the terminals produces a different series-parallel graph). Subsequently, we show how to adapt these oriented algorithms to the case of semioriented series-parallel graphs, where we still have a set of two distinguished terminals but neither has been designated as a source or sink. Finally, we discuss the case of unoriented series-parallel graphs, where no terminals have been distinguished and present a few observations and open questions relating to them. The algorithms we present generate the nonequivalent spanning trees of $G$ but never explicitly compute the automorphism group of $G$, revealing how the recursive structure of $G$'s automorphism group mirrors that of its spanning trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13480 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generating the Spanning Trees of Series-Parallel Graphs up to Graph Automorphism Karamchedu, Mithra Bang, Lucas Data Structures and Algorithms Combinatorics In this paper, we investigate the problem of generating the spanning trees of a graph $G$ up to the automorphisms or "symmetries" of $G$. After introducing and surveying this problem for general input graphs, we present algorithms that fully solve the case of series-parallel graphs, under two standard definitions. We first show how to generate the nonequivalent spanning trees of a oriented series-parallel graph $G$ in output-linear time, where both terminals of $G$ have been individually distinguished (i.e. applying an automorphism that exchanges the terminals produces a different series-parallel graph). Subsequently, we show how to adapt these oriented algorithms to the case of semioriented series-parallel graphs, where we still have a set of two distinguished terminals but neither has been designated as a source or sink. Finally, we discuss the case of unoriented series-parallel graphs, where no terminals have been distinguished and present a few observations and open questions relating to them. The algorithms we present generate the nonequivalent spanning trees of $G$ but never explicitly compute the automorphism group of $G$, revealing how the recursive structure of $G$'s automorphism group mirrors that of its spanning trees. |
| title | Generating the Spanning Trees of Series-Parallel Graphs up to Graph Automorphism |
| topic | Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2508.13480 |