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1、有向图字典式积的双控制数（英文）摘要令Y*(D)表示有向?DD的双控制数，Dm[Dn]表示有向图Dm和Dn的字典式职，其中Dm,Dn的阶数m，n分别大于等于2.本文首先给出Dm[Dn]的双控制数的上下界，然后确定如下有向图的双控制数的确切值：Dm[Kn];Km[Dn]；Kml，m2，…，mt[Dn];Cm[Dn];Pm[Cn]及Pm[Pn].关键词双控制数；字典式积；有向图LetD=(V(D)，A(D))beafinitedigraphwithoutloopsandmultiplearcswhereV(D)isthevertexsetandA(D)isthearcset.Foravert
2、exvEV(D)，N+D(v)andN~D(v)denotethesetsofoutneighborsandinneighbors,respectively,ofv,d+D(v)=
3、N+D(v)
4、andd-D(v)=
5、N~D(v)
6、denotetheoutdegreeandindegreeofvinD,respectively.AdigraphDisrregularifd+D(v)=d-D(v)=rforanyvertexvinD.GiventwoverticesuandvinD,wcsaythatuoutdominatesvifu=voruvEA(D),andvindominatesu
7、ifu=voruvEA(D).LetN+D[v]=N+D(v)U{v}.AvertexvoutdominatesallverticesinN+D[v]•八setSV(D)isanoutdominatingsetofDifSoutdominatesallverticesinV(D).TheoutdominationnumberofD,denotedbyy+(D),istheminimumcardinalityofanoutdominatingsetofD.Inthesameway,theindominationnumbery-(D)istheminimumcardinalityofanin
8、dominatingsetofD.Someresultsoftwindominationindigraphshavebeenobtainedin[1〜6].AsetSV(D)isatwindominatingsetofDifeachvertexofDiseitherinSorbothanoutneighbourofsomevertexinSandaninneighbourofsome(possiblydistinct)vertexinS.ThetwindominationnumberofD，denotedbyy*(D),istheminimumcardinalityofatwindomi
9、natingsetofD.Clearly,ifSisatwindominatingset，thenSisanoutdominatingsetandindominatingsetofD,thus丫*(D)y+(D)^1andy*(D)y-(D)》1.LetDm=(V(Dm),A(Dm))andDn=(V(Dn),A(Dn))betwodigraphswhichhavedisjointvertexsetsV(Dm)={x0，xl，…，xm-1}andV(Dn)={y0,yl,…，yn~l}anddisjointarcsetsA(Dm)andA(Dn),respectively.Thelexi
10、cographicproductDm[Dn]ofDmandDnhasvertexsetV(Dm)XV(Dn)={(xi,yj)
11、xiEV(Dm),yjEVn}and(xi,yj)(xi’，yj')EA(Dm[Dn])ifoneofthefollowingholds：(a)xixi’EA(Dm);(b)xi=xirandyjyj'EA(Dn).ForanyfixedvertexyjEV(Dn),thesubdigraphDyjmofDm[Dn]hasvertexsetV(Dyjm)={(xi,yj):foranyxiV(Dm)}，andarcsetA(Dyjm)={(xi,yj)(xi',
12、yj):xixi'EA(Dm)}.ItisclearthatDyjmDm.Similarly,foranyfixedvertexxiV(Dm),thesubdigraphDxinofDm[Dn]hasvertexsetV(Dxin)={(xi,yj):foranyyjV(Dn)},andarcsetA(Dxin)={(xi,yj)(xi,yj')：yjyjzA(Dn)}•ItisclearthatDxinDn.Inrecentyea
