MA Hong-xia, LIU Juan

(College of Mathematical Sciences, Xinjiang Normal University, Urumqi 830054, China)

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The Twin Domination Number of Lexicographic Product of Digraphs

MA Hong-xia, LIU Juan*

(College of Mathematical Sciences, Xinjiang Normal University, Urumqi 830054, China)

Letγ*(D) denote the twin domination number of digraphDand letDm[Dn] denote the lexicographic product ofDmandDn, digraphs of orderm,n≥2. In this paper, we first give the upper and lower bound of the twin domination number ofDm[Dn], and then determine the exact values of the twin domination number of digraphs:Dm[Kn];Km[Dn];Km1,m2,…,mt[Dn];Cm[Dn];Pm[Cn] andPm[Pn].

twin domination number; lexicographic product; digraphs

LetDm=(V(Dm)，A(Dm))andDn=(V(Dn)，A(Dn))betwodigraphswhichhavedisjointvertexsetsV(Dm)={x0，x1，…，xm-1}andV(Dn)={y0，y1，…，yn-1}anddisjointarcsetsA(Dm)andA(Dn),respectively.ThelexicographicproductDm[Dn]ofDmandDnhasvertexsetV(Dm)×V(Dn)={(xi,yj)|xi∈V(Dm),yj∈ Vn}and(xi,yj)(xi′,yj′)∈A(Dm[Dn])ifoneofthefollowingholds:(a) xixi′∈A(Dm);(b) xi=xi′andyjyj′∈A(Dn).

Inrecentyears,thedominationnumberofsomedigraphshasbeeninvestigatedin[7～16].However,todatefewresearchabouttwindominationnumberhasbeendoneforlexicographicproductofdigraphs.Inthispaper,westudythetwindominationnumberofDm[Dn].Firstly,wegivetheupperandlowerboundofthetwindominationnumberofDm[Dn],andthendeterminetheexactvaluesofthetwindominationnumberofdigraphs: Dm[Kn];Km[Dn];Km1,m2;…;mt[Dn]; Cm[Dn]; Pm[Cn]andPm[Pn].

1 Main results

Lemma 1 For anym,n≥2, thenγ*(Dm)≤γ*(Dm[Dn])≤γ*(Dm)γ*(Dn).

Now we prove thatγ*(Dm)γ*(Dn)≥γ*(Dm[Dn]). LetS1={xi1,…,xit} be a minimum twin dominating set ofDm. LetS2={yj1,…,yjw} be a minimum twin dominating set ofDn. SetS′={(xik,yjl)|xik∈S1,yjl∈S2}. For any vertex (xi,yj)∈V(Dm[Dn]), ifxi∈S1andyj∈S2, then (xi,yj)∈S′. Ifxi∈S1andyj?S2, there must existyjl,yjl′∈S2, such thatyjlyj,yjyjl′∈A(Dn). Since (xi,yjl)(xi,yj),(xi,yj)(xi,yjl′)∈A(Dm[Dn]) and (xi,yjl),(xi,yjl′)∈S′, (xi,yj) is twin dominated byS′. Ifxi?S1, there must existxik,xik′∈S1such thatxikxi,xixik′∈A(Dm), and there must existyr∈S2such that (xik,yr)(xi,yj),(xi,yj)(xik′,yr)∈A(Dm[Dn]) and (xik,yr), (xik′,yr)∈S′, so (xi,yj) is twin dominated byS′.

Therefore,S′ is a twin dominating set ofDm[Dn] and |S′|=|S1||S2|=γ*(Dm)γ*(Dn). Thus,γ*(Dm[Dn]) ≤γ*(Dm)γ*(Dn).

Clearly, the following theorem is obtained from Lemma 1.

Theorem 1 For anym,n≥2, thenγ*(Dm[Dn])=γ*(Dm) if and only ifγ*(Dn)=1.

According to Theorem 1, the following theorem is obvious.

Theorem 2 For anym,n≥2, thenγ*(Dm[Kn])=γ*(Dm).

Theorem 3 For any complete digraphKmand digraphDn, then

Proof By Lemma 1 and Theorem 1, it is clear thatγ*(Km[Dn])=1 ifγ*(Dn)=1. It is easy to see thatS={(x0,y0)} is a twin dominating set ofKm[Dn], where {y0} is a twin dominating set ofDn.

LetKm1,m2,…,mtbe the completet-partite digraphs. The next Theorem is obtained by Lemma 1 and Theorem 3.

Theorem 4 Letmi≥1, for anyi∈{1,2,…,t}. Then

WeemphasizethatverticesofadirectedcycleCnarealwaysdenotedbytheintegers{0,1,…,n-1},consideringmodulon,throughoutthispaper.ThereisanarcxyfromxtoyinCnifandonlyify≡x+1(modn).Forthefollowingaddition,wealwaysconsidermodulon.

Theorem 5 Form,n≥2, we have

Nowweconsidertwocases.

Case1γ+(Dn)=γ-(Dn)=1andγ*(Dn)≥2.

Let{yj1}and{yj2}beout-dominatingsetandin-dominatingsetofDn,respectively.Set

S3={(m-1,yj2)}.

Case2γ*(Dn)≥2andγ+(Dn)≥2orγ-(Dn)≥2.

Withoutlossofgenerality,weassumethatγ+(Dn)≥2.Itfollowsthatγ-(Dn)≥1,thatisγ+(Dn)≥γ-(Dn).Inthiscase,letJ={i|i∈{0,1,…,m-1}}suchthatbi=0andletJ′={i|i-1∈J}.ItiseasytoseethatJ∩J′=?.

Forcovenience,letPmbeadirectedpathwithvertexset{0,1,…,m-1}inthesequel.

Theorem6Letm,n≥2,then

Next,weconsidern≥3.Forconvenience,wefirstcountthenumberofbifromi=1toi=m-2.Therearetwocasestobeconsidered:

Case1bm-2≠0.

Then

Case2bm-2=0 (thatis|J|≥1).Weconsiderthefollowingtwosubcases:

Itfollowsthattheremustexistsomeevenintegerjsuchthatbm-j≠0.Then

S3={(i,0)|i∈{1,2，…，m-2}}.

ThefollowingCorollaryisobtainedfromTheorem6.

Corollary1Letm,n≥2,then

ProofTheframeworkandmainideaforthisproofarethesameasthatofTheorem6.Thepointtobemodehereisthatforthecasem≠3,oneshouldconsiderfoursubcases.n=2,3,4andn≥5.Hence,theproofisleftforthereaders.

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(編輯 HWJ)

2016-05-03

國(guó)家自然科學(xué)基金資助項(xiàng)目(11301450,61363020，11226294)；新疆自治區(qū)青年科技創(chuàng)新人才培養(yǎng)工程資助項(xiàng)目(2014731003)

有向圖字典式積的雙控制數(shù)

馬紅霞，劉 娟*

(新疆師范大學(xué)數(shù)學(xué)科學(xué)學(xué)院，中國(guó) 烏魯木齊 830054)

令γ*(D)表示有向圖D的雙控制數(shù)，Dm[Dn]表示有向圖Dm和Dn的字典式積，其中Dm,Dn的階數(shù)m,n分別大于等于2.本文首先給出Dm[Dn]的雙控制數(shù)的上下界，然后確定如下有向圖的雙控制數(shù)的確切值：Dm[Kn];Km[Dn];Km1,m2,…,mt[Dn];Cm[Dn];Pm[Cn]及Pm[Pn].

雙控制數(shù)；字典式積；有向圖

10.7612/j.issn.1000-2537.2016.06.014

*通訊作者,E-mail：liujuan1999@126.com