Exact Total-Domination Embeddings and Computational Hardness in δ-Complement Graphs
Year: 2026
Type: preprint
Abstract: No.55 preprint v1.0, 5 November 2026. Ryutaro Yonezu — Independent Researcher. We give a polynomial-time construction F for any isolate-free finite simple graph Q on m>=2 vertices with gamma_t(F(Q))=3m, gamma_t(F(Q)_delta)=2m+gamma_t(Q), and k(F(Q))=m+1, where k counts distinct original vertex degrees. The construction preserves bipartiteness and trees. A self-contained Set Cover reduction proves NP-completeness of total domination in the delta-complement of a BIPARTITE INPUT graph. The output delta-complement need not be bipartite. A tree family has unbounded excess over the separate auxiliary bound 2k; Chen's original total-domination SUM conjecture is neither proved nor refuted. Independent AI audit: PASS within the audited scope, based on user-provided mathematical and v0.2 documentation-only differential-audit reports. This is not human peer review or a new independent mathematical audit performed during publication. Novelty: INCONCLUSIVE. Human peer review: NOT DONE / Not human peer-reviewed. References: PARTIAL; unavailable original full texts are not claimed read. The reproducibility ZIP preserves the original v0.1/v0.2 archives and all seven frozen proof/code/evidence/script files byte-for-byte. It includes the complete v1.0 manuscript, README, publication status and editorial diff. Historical HOLD/NOT DONE labels remain archival snapshots. Current author authorization and audit attribution are explained separately. Windows reproduction: the user-supplied original transcript records Windows 10 / PowerShell 5.1 / Python 3.10.11 / NumPy 2.2.6 / SciPy 1.15.3 / NetworkX 3.4.2 on 2026-11-05 14:11 JST: 16 MILP, 33 Atlas, 40 Set Cover and 11 composed F(B) cases PASS, with the original v0.1 ZIP SHA256 identical before and after execution. This is a v0.1 source run using the separate No55_Run_Windows_v012.ps1 hotfix, not Windows execution of v0.2 or v1.0. The original transcript and hotfix are included; separately generated Windows JSON outputs were not supplied and are not claimed included. Copyright (c) 2026 Ryutaro Yonezu. All rights reserved, including manuscript, code and supplementary research materials. GitHub repository: https://github.com/yonezaemon1-hub/no55-delta-complement-total-dominationGitHub Release v1.0: https://github.com/yonezaemon1-hub/no55-delta-complement-total-domination/releases/tag/v1.0 SHA256 PDF: 7a455ac2dccc74fc9c6ae00a893834eabeb9fc256b016910ca75f344b6775deeSHA256 reproducibility ZIP: 4216c450f60c768f3d8ff6b0b10a17364493d3e2a1d7625edcf788943153ad20
Author: Yonezu, Ryutaro
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225343136