Identifying bubble-like subgraphs in linear-time via a unified SPQR-tree framework
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| Main Authors: | , , , , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908949386625024 |
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| author | Sena, Francisco Politov, Aleksandr Moumard, Corentin Cairo, Massimo Rizzi, Romeo Cáceres, Manuel Schmidt, Sebastian Harviainen, Juha Tomescu, Alexandru I. |
| author_facet | Sena, Francisco Politov, Aleksandr Moumard, Corentin Cairo, Massimo Rizzi, Romeo Cáceres, Manuel Schmidt, Sebastian Harviainen, Juha Tomescu, Alexandru I. |
| contents | A fundamental algorithmic problem in computational biology is to find all subgraphs of a given type (superbubbles, snarls, and ultrabubbles) in a directed or bidirected input graph. These correspond to regions of genetic variation and are useful in analyzing collections of genomes. We present the first linear-time algorithms for identifying all snarls and all ultrabubbles, resolving problems open since 2018. The algorithm for snarls relies on a new linear-size representation of all snarls with respect to the number of vertices in the graph. We employ the well-known SPQR-tree decomposition, which encodes all 2-separators of a biconnected graph. After several dynamic-programming-style traversals of this tree, we maintain key properties (such as acyclicity) that allow us to decide whether a given 2-separator defines a subgraph to be reported. A crucial ingredient for linear-time complexity is that acyclicity of linearly many subgraphs can be tested simultaneously via the problem of computing all arcs in a directed graph whose removal renders it acyclic (so-called feedback arcs). As such, we prove a fundamental result for bidirected graphs, that may be of independent interest: all feedback arcs can be computed in linear time for tipless bidirected graphs, while in general this is at least as hard as matrix multiplication, assuming the k-Clique Conjecture. Our results form a unified framework that also yields a completely different linear-time algorithm for finding all superbubbles. Although some of the results are technically involved, the underlying ideas are conceptually simple, and may extend to other bubble-like subgraphs. More broadly, our work contributes to the theoretical foundations of computational biology and advances a growing line of research that uses SPQR-tree decompositions as a general tool for designing efficient algorithms, beyond their traditional role in graph drawing. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_08071 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Identifying bubble-like subgraphs in linear-time via a unified SPQR-tree framework Sena, Francisco Politov, Aleksandr Moumard, Corentin Cairo, Massimo Rizzi, Romeo Cáceres, Manuel Schmidt, Sebastian Harviainen, Juha Tomescu, Alexandru I. Data Structures and Algorithms A fundamental algorithmic problem in computational biology is to find all subgraphs of a given type (superbubbles, snarls, and ultrabubbles) in a directed or bidirected input graph. These correspond to regions of genetic variation and are useful in analyzing collections of genomes. We present the first linear-time algorithms for identifying all snarls and all ultrabubbles, resolving problems open since 2018. The algorithm for snarls relies on a new linear-size representation of all snarls with respect to the number of vertices in the graph. We employ the well-known SPQR-tree decomposition, which encodes all 2-separators of a biconnected graph. After several dynamic-programming-style traversals of this tree, we maintain key properties (such as acyclicity) that allow us to decide whether a given 2-separator defines a subgraph to be reported. A crucial ingredient for linear-time complexity is that acyclicity of linearly many subgraphs can be tested simultaneously via the problem of computing all arcs in a directed graph whose removal renders it acyclic (so-called feedback arcs). As such, we prove a fundamental result for bidirected graphs, that may be of independent interest: all feedback arcs can be computed in linear time for tipless bidirected graphs, while in general this is at least as hard as matrix multiplication, assuming the k-Clique Conjecture. Our results form a unified framework that also yields a completely different linear-time algorithm for finding all superbubbles. Although some of the results are technically involved, the underlying ideas are conceptually simple, and may extend to other bubble-like subgraphs. More broadly, our work contributes to the theoretical foundations of computational biology and advances a growing line of research that uses SPQR-tree decompositions as a general tool for designing efficient algorithms, beyond their traditional role in graph drawing. |
| title | Identifying bubble-like subgraphs in linear-time via a unified SPQR-tree framework |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2604.08071 |