Twin Domination in Digraphs.
Bibliographic record
Abstract
TwinDominationinDigraphsGaryChartrand,estern Whigan Micy ersit Univeter Pelmann, Danky ersit UnivofNatal,Durbanhelle Micultz, h Scy ersit Univofada, NevLasegas VHendaC.art, Swy ersit UnivofNatal,DurbanAbstractAertex vvinadigraphDout-dominatesitselfasell wallertices vuh sucthat(v,)isanarcofD;whilein-dominatesb othitselfandallertices vwh sucthat(,v)isanarcofD.AsetSawin tdominatingsetofDifery evertex visout-dominatedy bsomeofSandin-dominatedy bsomeertex v.Theum minimy cardinalitawin tdominatingsetisthedominationb umer n(D)of.Itwn shothat(D)2p=3forery evdigraphoforderpving hanoertex vofin-degree0orout-degree0.er, v Moreoe we givaNordhaus-Gaddumyp teb oundfor,ande transitivdigraphse we givasharpupp erb oundforthewin tdominationb umer nintermsoforderandum minimdegree.or FagraphG,theupp ertable orienwin tdominationb umer nD O M()istheum maximwin tdominationb umer n(D)er v oalltations orienofG;whiletheer w lotable orienwin tdominationb umer ndom()istheum minimh sucwin tdominationb umer. nItwn shothatforh eacgraphGandteger incwithdom()D O M),thereexistsantation orienDofGh sucthat()=c.1tro InductionDominationingraphshasb eenstudiedely extensivt recenears. yTheb o oky bynes, HaHedetniemi,andSlater[7]istirely enoted devtothisarea.Dominationind cte e dirgraphs,er, ev w hohasnotattractedthesamet amounoftion atteny bhers researc(seetheey surv[6]),exceptforstudyofernels kdigraphs.Anaturaldirectedanaloguetothedominationb umer nofanundirectedgraphistheout-dominationb umer nofadigraphD,de nedasum minimy cardinalitofasubsetStheertex vsetDh sucthath eaciseitherinoranb out-neighourofsomeertex vinS.Theout-dominationb umer nandthein-dominationb umer, nh whicisde ned, analogouslyarestudiede.g.in[5].Inthispap ere wprop oseathird,equallynaturaldirectedanaloguetodominationingraphs.Thetwindominationer numbofadigraphDistheum minimy cardinalitofasubsetStheertex vsetDh sucthath eaciseitherinorb oth,anb out-neighourofsomeertex vinSandb in-neighour(p ossiblydistinct)ertex vinS.or Fterminologyandnotationp ertainingtodigraphs,e wreferto[2].1
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.012 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".