Twin Domination in Digraphs.
Gary Chartrand, Peter Dankelmann, Michelle Schultz, Henda C. Swart · Ars Combinatoria · 2003
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