{"paper":{"title":"Connectivity-Preserving Important Separators: A Framework for Cut-Uncut Problems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Connectivity-preserving important separators can be enumerated in 2^{O(k log k)} time, extending classical separator techniques to cut-uncut problems that require both disconnection and internal connectivity preservation.","cross_cats":["cs.CC"],"primary_cat":"cs.DS","authors_text":"Batya Kenig","submitted_at":"2025-11-19T20:13:23Z","abstract_excerpt":"Important separators are a cornerstone of parameterized algorithms for graph separation: they reduce an a priori enormous search space of separators to a small, structured family that can be enumerated efficiently. This principle has been remarkably successful for parameterized separation problems, but it does not address cut-uncut problems, where one must cut some connections while preserving the connectivity of a given set of terminals. These connectivity-preservation requirements create a qualitatively different type of structure, and the classical important-separator machinery no longer gi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Our main result shows that this family is highly structured: the number of connectivity-preserving important separators of size at most k is 2^{O(k log k)}, and they can be enumerated within the same bound up to polynomial factors.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The framework assumes that the connectivity constraints (equivalence classes of terminals that must remain connected) are provided explicitly and that the underlying graph is undirected and simple; if the constraints are implicit or the graph has directed edges or weights, the enumeration bound may not apply directly.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Connectivity-preserving important separators of size at most k number 2^{O(k log k)} and can be enumerated in the same bound, yielding 2^{O(k log k)} FPT time for constant-class Node Multiway Cut-Uncut.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Connectivity-preserving important separators can be enumerated in 2^{O(k log k)} time, extending classical separator techniques to cut-uncut problems that require both disconnection and internal connectivity preservation.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"70bd85ca11044a0a448d7ca4409cc1cea9adff04d4a64f166a08cdc055fce93a"},"source":{"id":"2511.15849","kind":"arxiv","version":4},"verdict":{"id":"96a8f257-af78-457c-8a13-e925f46acc3b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-17T20:02:05.479930Z","strongest_claim":"Our main result shows that this family is highly structured: the number of connectivity-preserving important separators of size at most k is 2^{O(k log k)}, and they can be enumerated within the same bound up to polynomial factors.","one_line_summary":"Connectivity-preserving important separators of size at most k number 2^{O(k log k)} and can be enumerated in the same bound, yielding 2^{O(k log k)} FPT time for constant-class Node Multiway Cut-Uncut.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The framework assumes that the connectivity constraints (equivalence classes of terminals that must remain connected) are provided explicitly and that the underlying graph is undirected and simple; if the constraints are implicit or the graph has directed edges or weights, the enumeration bound may not apply directly.","pith_extraction_headline":"Connectivity-preserving important separators can be enumerated in 2^{O(k log k)} time, extending classical separator techniques to cut-uncut problems that require both disconnection and internal connectivity preservation."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2511.15849/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}