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The task is to decide if there exists a set $X\\subseteq V(G)$ such that the weight of $X$ plus the weight of a heaviest component of $G-X$ is at most $p$. Among other results, we prove that:\n  (1) wVI is NP-complete on co-comparability graphs, even if each vertex has weight $1$;\n  (2) wVI can be solved in $O(p^{p+1}n)$ time;\n  (3) wVI admits a kernel with at most $p^3$ vertices.\n  Result (1) refutes a conjecture by Ray and Deogun and answers an open q","authors_text":"Markus Sortland Dregi, P{\\aa}l Gr{\\o}n{\\aa}s Drange, Pim van 't Hof","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-03-25T13:11:09Z","title":"On the Computational Complexity of Vertex Integrity and Component Order Connectivity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.6331","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b865f7541938e39c6e404a16e6d60dda59e6a1613877daab0496f5feb942511c","target":"record","created_at":"2026-05-18T02:32:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"95f35d975d6e6ae163ee09d20ecbd4e7ab42ac9e5b0dcc55e53bca61a534a695","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-03-25T13:11:09Z","title_canon_sha256":"0dc6baaa7bdf9889058a32cfe0d95341d9ed9e16e5cd39f0d0972c21c605166d"},"schema_version":"1.0","source":{"id":"1403.6331","kind":"arxiv","version":2}},"canonical_sha256":"9a862dfc2665e5752b7a002c067d5f85fd4bed5c9e34ae6b245b5c1c255b6da1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9a862dfc2665e5752b7a002c067d5f85fd4bed5c9e34ae6b245b5c1c255b6da1","first_computed_at":"2026-05-18T02:32:07.521030Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:07.521030Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CsleePz0EsfhTZQeJo/hRBNLKRW621SbST0Mnodx7Bsu3OlNadANmp172qNmwdmeGKWz5sXj3JVoWMPeI6M8BA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:07.521430Z","signed_message":"canonical_sha256_bytes"},"source_id":"1403.6331","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b865f7541938e39c6e404a16e6d60dda59e6a1613877daab0496f5feb942511c","sha256:7d5099d934de4e68b9687f292060e3de9198422449d4cde67eee37ab529d3362"],"state_sha256":"7844ae108190cfd4c3a66e8625fbe2ef5a122c41e0a02cfcbbc7c79818dbb71b"}