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The minimum number of colors in such a coloring is the strong conflict-free vertex-connection number $\\operatorname{svcfc}(G)$. We study this problem under the parameter twin cover.\n  Let $X$ be a twin cover of $G$ of size $t$, and let $k$ be the target number of colors. In our first result, given $(G,k)$ together with a twin cover $X$, we reduce in polynomial time to an equivalent annotated instan"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.13299","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2026-05-13T10:13:01Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"e401756958e668504b3b09e20c563c1d223ae8a52eb306e247e1a512b7c22894","abstract_canon_sha256":"cf35c419ebe748131ac7a76d0f046512d08b4b1f6d53d83c04ba7ac6a5ac939a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:49.058583Z","signature_b64":"1KCm7+Wzbs1Us9GWyXEJNmMJj6ZgRu4G7qPWotH8fFoeyPDmTBX21ng8yzhlx+/PhxPPQiF260MiECAMjDPABA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8d11efc51625d49bd9e53ccfb1f6da56358fb8e061b32caaaf0573a6b0622b10","last_reissued_at":"2026-05-18T02:44:49.058184Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:49.058184Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Strong Conflict-Free Vertex-Connection via Twin Cover: Kernelization and Chromatic Bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A graph with twin cover of size t has strong conflict-free vertex-connection number at most its chromatic number plus t.","cross_cats":["cs.DS"],"primary_cat":"cs.DM","authors_text":"Samuel German","submitted_at":"2026-05-13T10:13:01Z","abstract_excerpt":"A vertex-coloring of a connected graph $G$ is a strong conflict-free vertex-connection coloring if every two distinct vertices are joined by a shortest path on which some color appears exactly once. The minimum number of colors in such a coloring is the strong conflict-free vertex-connection number $\\operatorname{svcfc}(G)$. We study this problem under the parameter twin cover.\n  Let $X$ be a twin cover of $G$ of size $t$, and let $k$ be the target number of colors. In our first result, given $(G,k)$ together with a twin cover $X$, we reduce in polynomial time to an equivalent annotated instan"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Given (G,k) together with a twin cover X of size t, we reduce in polynomial time to an equivalent annotated instance on at most max{2,t+(t+1)k 2^{t+k-1}} vertices. Every connected graph G with twin cover X of size t satisfies χ(G) ≤ svcfc(G) ≤ χ(G) + t.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The kernelization result assumes a twin cover X is supplied as part of the input; the FPT claim for tc(G) + k therefore depends on either the cover being given or on the complexity of computing it separately.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Kernelization establishes FPT for strong CFVC number by twin cover plus k, with svcfc(G) bounded between χ(G) and χ(G) plus twin cover size.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A graph with twin cover of size t has strong conflict-free vertex-connection number at most its chromatic number plus t.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5b766251d78386c90c3028ae89282a95889195e111b68e4bfd27fec2f999fcf3"},"source":{"id":"2605.13299","kind":"arxiv","version":1},"verdict":{"id":"11e280b8-7e99-4e73-9bc7-47d1c235229b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:02:45.963381Z","strongest_claim":"Given (G,k) together with a twin cover X of size t, we reduce in polynomial time to an equivalent annotated instance on at most max{2,t+(t+1)k 2^{t+k-1}} vertices. Every connected graph G with twin cover X of size t satisfies χ(G) ≤ svcfc(G) ≤ χ(G) + t.","one_line_summary":"Kernelization establishes FPT for strong CFVC number by twin cover plus k, with svcfc(G) bounded between χ(G) and χ(G) plus twin cover size.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The kernelization result assumes a twin cover X is supplied as part of the input; the FPT claim for tc(G) + k therefore depends on either the cover being given or on the complexity of computing it separately.","pith_extraction_headline":"A graph with twin cover of size t has strong conflict-free vertex-connection number at most its chromatic number plus t."},"references":{"count":11,"sample":[{"doi":"10.1016/j.dam.2024.03.021","year":2024,"title":"Discrete Applied Mathematics352, 88–104 (2024)","work_id":"7512394d-3307-4727-b596-0b1f2d9dfbab","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.7151/dmgt.2036","year":2018,"title":"Discussiones Mathematicae Graph Theory38(4), 911–920 (2018)","work_id":"80096e22-90d9-44fe-bd92-681d0555cdd0","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1137/s0097539702431840","year":2003,"title":"SIAM Journal on Computing33(1), 94–136 (2003)","work_id":"0895d1b1-b83f-4305-9bb9-ccab69387e97","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.dam.2025.12.025","year":2026,"title":"Discrete Applied Mathematics383, 85–93 (2026)","work_id":"ed16aaec-6e9d-4626-9bb4-b6d44b7f2641","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.46298/dmtcs.2136","year":2015,"title":"Discrete Mathematics & Theoretical Computer Science17(2), 77–100 (2015)","work_id":"e05ab08c-e76c-427e-ae89-5c01ae5c0f81","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":11,"snapshot_sha256":"dcfdefb5c8c12d3a4cec0e3c097d6a1982e061f4b617d34d9376f60035239f2d","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"b48e3de948d75832b114bc363b10a8cfaad3c0a185d1a37bc738b4f47f624648"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.13299","created_at":"2026-05-18T02:44:49.058243+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.13299v1","created_at":"2026-05-18T02:44:49.058243+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13299","created_at":"2026-05-18T02:44:49.058243+00:00"},{"alias_kind":"pith_short_12","alias_value":"RUI67RIWEXKJ","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"RUI67RIWEXKJXWPF","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"RUI67RIW","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":1,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RUI67RIWEXKJXWPFHTH3D5W2KY","json":"https://pith.science/pith/RUI67RIWEXKJXWPFHTH3D5W2KY.json","graph_json":"https://pith.science/api/pith-number/RUI67RIWEXKJXWPFHTH3D5W2KY/graph.json","events_json":"https://pith.science/api/pith-number/RUI67RIWEXKJXWPFHTH3D5W2KY/events.json","paper":"https://pith.science/paper/RUI67RIW"},"agent_actions":{"view_html":"https://pith.science/pith/RUI67RIWEXKJXWPFHTH3D5W2KY","download_json":"https://pith.science/pith/RUI67RIWEXKJXWPFHTH3D5W2KY.json","view_paper":"https://pith.science/paper/RUI67RIW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.13299&json=true","fetch_graph":"https://pith.science/api/pith-number/RUI67RIWEXKJXWPFHTH3D5W2KY/graph.json","fetch_events":"https://pith.science/api/pith-number/RUI67RIWEXKJXWPFHTH3D5W2KY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RUI67RIWEXKJXWPFHTH3D5W2KY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RUI67RIWEXKJXWPFHTH3D5W2KY/action/storage_attestation","attest_author":"https://pith.science/pith/RUI67RIWEXKJXWPFHTH3D5W2KY/action/author_attestation","sign_citation":"https://pith.science/pith/RUI67RIWEXKJXWPFHTH3D5W2KY/action/citation_signature","submit_replication":"https://pith.science/pith/RUI67RIWEXKJXWPFHTH3D5W2KY/action/replication_record"}},"created_at":"2026-05-18T02:44:49.058243+00:00","updated_at":"2026-05-18T02:44:49.058243+00:00"}