Efficient reversal of transductions of sparse graph classes
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909996823871488 |
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| author | Dreier, Jan Gajarský, Jakub Pilipczuk, Michał |
| author_facet | Dreier, Jan Gajarský, Jakub Pilipczuk, Michał |
| contents | (First-order) transductions are a basic notion capturing graph modifications that can be described in first-order logic. In this work, we propose an efficient algorithmic method to approximately reverse the application of a transduction, assuming the source graph is sparse. Precisely, for any graph class $\mathcal{C}$ that has structurally bounded expansion (i.e., can be transduced from a class of bounded expansion), we give an $O(n^4)$-time algorithm that given a graph $G\in \mathcal{C}$, computes a vertex-colored graph $H$ such that $G$ can be recovered from $H$ using a first-order interpretation and $H$ belongs to a graph class $\mathcal{D}$ of bounded expansion. This answers an open problem raised by Gajarský et al. In fact, for our procedure to work we only need to assume that $\mathcal{C}$ is monadically stable (i.e., does not transduce the class of all half-graphs) and has inherently linear neighborhood complexity (i.e., the neighborhood complexity is linear in all graph classes transducible from $\mathcal{C}$). This renders the conclusion that the graph classes satisfying these two properties coincide with classes of structurally bounded expansion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_14906 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Efficient reversal of transductions of sparse graph classes Dreier, Jan Gajarský, Jakub Pilipczuk, Michał Logic in Computer Science Discrete Mathematics (First-order) transductions are a basic notion capturing graph modifications that can be described in first-order logic. In this work, we propose an efficient algorithmic method to approximately reverse the application of a transduction, assuming the source graph is sparse. Precisely, for any graph class $\mathcal{C}$ that has structurally bounded expansion (i.e., can be transduced from a class of bounded expansion), we give an $O(n^4)$-time algorithm that given a graph $G\in \mathcal{C}$, computes a vertex-colored graph $H$ such that $G$ can be recovered from $H$ using a first-order interpretation and $H$ belongs to a graph class $\mathcal{D}$ of bounded expansion. This answers an open problem raised by Gajarský et al. In fact, for our procedure to work we only need to assume that $\mathcal{C}$ is monadically stable (i.e., does not transduce the class of all half-graphs) and has inherently linear neighborhood complexity (i.e., the neighborhood complexity is linear in all graph classes transducible from $\mathcal{C}$). This renders the conclusion that the graph classes satisfying these two properties coincide with classes of structurally bounded expansion. |
| title | Efficient reversal of transductions of sparse graph classes |
| topic | Logic in Computer Science Discrete Mathematics |
| url | https://arxiv.org/abs/2601.14906 |