Strong Efficient Domination in Graphs

Transcription

1 P P P IJISET - Internatonal Journal of Innovatve Scence, Engneerng & Technology, Vol Iue 4, June 04 wwwjetcom Strong Effcent Domnaton n Graph ISSN NMeenaP P, ASubramananP P, VSwamnathanP PDepartment of Mathematc, The MDT Hndu College, Trunelvel 67 00, Tamlnadu, Inda PDean, College Development Councl, Manonmanam Sundaranar Unverty, Trunelvel 67 0, Tamlnadu, Inda 3 PRamanujan Reearch Center, Department of Mathematc, Sarawath Narayanan College, Madura 65 0, Tamlnadu, Inda Abtract Let G (V, E) be a mple graph E Sampathkumar and LPuhpalatha ntroduced the concept of trong and weak domnaton n graph [5] In th paper, th concept extended to effcent domnaton A ubet S of V called a trong (weak) effcent domnatng et of G f for every v V, NRR[v] S ( NRwR[v] S ), where NRR(v) { u V: uv E, deg(u) deg(v) }and NRwR(v) { u V : uv E, deg(v) deg(u) NRR[v] NRR(v) {v NRwR[v] NRwR(v) {v} The mnmum cardnalty of a trong (weak) effcent domnatng et called trong (weak) effcent domnaton number of G and denoted by γrerr R(γRweR) A graph trong effcent f there ext a trong effcent domnatng et In th paper we fnd clae of graph whch are trong effcent and compare trong effcent domnaton number wth γrr, ΓRR, RR and βrr Keyword: Strong effcent domnaton number, Full degree vertex, Strong and weak neghbour AMS Subject Clafcaton (00): 05C69 Introducton Throughout th paper, we conder fnte, undrected, mple graph Let G (V,E) be a graph The degree of any vertex u n G the number of edge ncdent wth u and denoted by deg u The mnmum and maxmum degree of a vertex denoted by δ and repectvely A vertex of degree 0 n G called an olated vertex and a vertex of degree called a pendant vertex For all graph theoretc termnologe and notaton, we follow Harary [4] The followng defnton are neceary for the preent tudy Defnton [3]: A ubet M of E called a matchng n G f t element are edge and no two are adjacent n G; the two end of an edge n M are ad to be matched under M A matchng M aturate a vertex v and v ad to be M-aturated, f ome edge of M ncdent wth v; otherwe v M-unaturated If every vertex of G M-aturated, then the matchng M called perfect Defnton[]: Generalzed Hajo graph, denoted by [KRnR] havng n + n vertce formed by takng KRnR, addng n new vertce and jonng each one of them to the end of exactly one edge of KRnR 3 Defnton[5] : A ubet S of V of a graph G called a domnatng et f every vertex n V \ S adjacent to a vertex n S The domnaton number γ the mnmum cardnalty of a domnatng et of G 4 Defnton[6] : A ubet S of V called a trong domnatng et of G f for every v V S there ext u S uch that u and v are adjacent and deg u deg v 5 Defnton[] : A ubet S of V called an effcent domnatng et of G f for every v V, N[v] S A et of pont ndependent f no two pont are adjacent A trong domnatng et of G whch alo ndependent called an ndependent trong domnatng et of G The mnmum cardnalty of the ndependent trong domnatng et of G called an ndependent trong domnaton number of G and denoted by RR The maxmum cardnalty of an ndependent trong domnatng et of G called the upper ndependent trong domnaton number of G and denoted by βrr Motvated by thee defnton, we have defned a new type of domnaton called trong effcent domnaton A graph G called a trong effcent graph f t admt a trong effcent domnatng et The mnmum cardnalty of a trong effcent domnatng et called the trong effcent 7

2 n SRR SRR SRR any IJISET - Internatonal Journal of Innovatve Scence, Engneerng & Technology, Vol Iue 4, June 04 domnaton number of G and denoted by γrer In th paper we clafy graph whch are trong effcent and we compare trong effcent domnaton number wth γrr, ΓRR, RR and βrr We have proved that there are graph n whch everal trong effcent domnatng et of dfferent cardnalte ext whch nteretng n the ene that all effcent domnatng et have the ame cardnalty γ Man Reult Defnton : Let G (V, E) be a mple graph A ubet S of V called a trong (weak) effcent domnatng et of G f for every v V, NRR[v] S ( NRwR[v] S ), where NRR(v) { u V : uv E, deg(u) deg(v) } and NRwR(v) { u V : uv E, deg(v) deg(u) NRR[v] NRR(v) {v} and NRwR[v] NRwR(v) {v} The mnmum cardnalty of a trong (weak) effcent domnatng et of G called the trong (weak) effcent domnaton number of G and denoted by γrerr R(γRweRR R) A graph G trong effcent f and only f there ext a trong effcent domnatng et of G Not all graph admt trong effcent domnatng et Example : Conder the followng fgure G Fgure : Clearly {vrr, vrr} a trong effcent domnatng et of G Therefore γrer Example 3: Graph wthout trong effcent domnatng et: Conder the graph GR Rand GRR the followng fgure and 3 repectvely Fgure : wwwjetcom ISSN Fgure 3: In GRR domnatng et (whch alo a trong domnatng et) not effcent In GRR, no trong domnatng et effcent Remark 4: γr R γre R For, Let S be a mnmum trong effcent domnatng et of G Let v V-S Then NRR [v] S e there ext u S, uch that u and v are adjacent and deg u deg v Therefore S a trong domnatng et of G ThereforeγR R γre R Theorem 5: For any path P RmR, n f m 3n, n N γrer(p RmR) n + f m 3n +, n N n + f m 3n +, n N Proof: Cae (): Let G PR3n R, n N Let vrr, vrr, vr3r vr3n Rbe the vertce of V (PR3nR) {vrr, vr5r, vr8r vr3n-r} the unque trong domnatng et of P R3nR It alo the trong effcent domnatng et of PR3nR Therefore γrer n Therefore γrer(pr3nr) n, for all n N Cae (): Let G PR3n+R, n N Let V {vrr, vrr, vr3r,,vr3nr,vr3n+r} {vrr, vr5r, vr8r vr3n-r, vr3n+r} and SRR {vrr, vr3r, vr6r, vr9r,,vr3nr} are two trong effcent domnatng et of G {vrr, vr5r, vr8r vr3n-r} {vr3n+r} SRR n + {vrr} {vr3r, vr6r, vr9r,,vr3nr} SRR n + Therefore γrer(pr3n+r) n +, for all n N Snce n + γrr(pr3n+r) γrer(pr3n+r) We ee that γrer(pr3n+r) n + Cae (): Let G PR3n+R, n N Let V {vrr, vrr, vr3r,,vr3nr,vr3n+r, vr3n+r} S {vrr, vr3r, vr6r, vr9r,,vr3nr, vr3n+r} a trong effcent domnatng et of G That S n + Therefore γrer(pr3n+r) n +, for all n N Snce n + γrr(pr3n+r) γrer(pr3n+r) We ee that γrer(pr3n+r) n + Example 6: Conder the path P R9 Fgure 4: 73

3 SRR RvR9R, Rn P P IJISET - Internatonal Journal of Innovatve Scence, Engneerng & Technology, Vol Iue 4, June 04 wwwjetcom ISSN {vrr, vr5r, vr8r} the trong effcent domnatng et of Example : For the graph G n the followng PR9R γre R(P R9R) 3 fgure Conder the path P R0 Fgure 5: {vrr, vr3r,vr6r, vr9r {vrr, vr5r, vr8r, vr0r} are two trong effcent domnatng et of PR0R γre R(PR0R) 4 Conder the path P R Fgure 6: { vrr, vr3r, vr6r,r vrr} the trong effcent domnatng et of PRR γre R(PRR) 5 Theorem 7: γrer(cr3nr) n, for all n N Proof: Let G CR3nR, n N Let V {vrr, vrr, vr3r,,vr3nr} {vrr, vr4r, vr7r,,vr3n-r SR R {vrr, vr5r, vr8r, vr3n-r} and SR3R {vr3r, vr6r, vr9r,,vr3nr} are the trong effcent domnatng et of G SRR SRR SR3R n γrer(cr3nr) n nce n γrr(cr3nr) γrer(cr3nr) we ee that γrer(cr3nr) n, for all n N Obervaton 8: γre R(KRnR), R N γre R(KRR,RnR) 3 γre R(WRn+R), n 3, n N Theorem 9: Every trong effcent domnatng et ndependent Proof: Let S be a trong effcent domnatng et Let u, v S Suppoe u and v are adjacent Let wthout lo of generalty, d(u) d(v) Then NRR[v] S, a contradcton Therefore S ndependent Theorem 0: If S a trong effcent domnatng et of a connected graph G then V S a domnatng et of G Proof: Snce every trong effcent domnatng et ndependent and G connected, every vertex n S adjacent to at leat one vertex V S Therefore V SP a domnatng et of G Remark : If S a trong effcent domnatng et of G then V SP Pneed not be a weak domnatng et of G Fgure 7: S {vrr,vr5r,vr6r,vr7r} a trong effcent domnatng et of G Snce vr5r,vr6r,vr7 Rare trongly domnated by the vertce of V S, V SP not a weak domnatng et of G Remark 3: () Every weak effcent domnatng et of a graph G a weak domnatng et of G () Complement of a trong effcent domnatng et of G need not be a weak effcent domnatng et of G(for example, n the graph, S {vrr,vr5r,vr6r,vr7r} trong effcent domnatng et of G and V SP P not a weak effcent domnatng et of G) Obervaton 4: Let G be a connected graph wth at leat pendant vertce and γrer Then G ha no perfect matchng (Snce γrer mple that G ha a full degree vertex) Theorem 5: Let G be a connected graph and V n (n even) If γrer > n/ then G ha no perfect matchng Proof: Let S be a γrer-et of a connected graph G and V n Let S t Snce γrer > n, t > n Suppoe that G ha a perfect matchng M Then every vertex of G M-aturated Therefore every vertex of S M-aturated Snce S ndependent, every lne of M ha ether one end n S and other end n V S or both end n V S Snce M a perfect matchng, M n But S t > n Therefore t n vertce of S are M-unaturated, a contradcton Therefore G ha no perfect matchng Remark 6: RR γre R βrr Remark 7: There ext graph n whch γr R < γre R For, Let G DRr,rR, r 74

4 βrr SRR SRR SR3R SR4R SR5R SR6R SR7R SR8R Rn N are Rn N are IJISET - Internatonal Journal of Innovatve Scence, Engneerng & Technology, Vol Iue 4, June 04 wwwjetcom ISSN γre R r + and γr R {vrr, vr4r, vr6r, vrr, vrr, vr3r, vr7r, vrr, vrr,r RvR3R SR4R 0 Remark 8: Gven any potve nteger k, there ext a connected graph G uch that γre R - γr R k {vrr, vr4r, vr7r, vr8r, vr9r, vr0r, vr3r, vr7r, vr9r,r RvR0R SR5R 0 { vrr, vr4r, vr6r, vr8r, vr9r, vr0r, vr3r, vr7r, vrr, vrr,r RvR3R Example 9: Let G DRk+,k+R, k γre R k + and γr R SR6R { vrr, vr5r, vr7r, vr8r, vr9r, vr0r, vr3r, vr5r, vr6r, vr9r, vr0r Remark 0: It clear that γr R RR SR7R, γre R βrr ΓRR SRR, to7 are ndependent domnatng et of G SR R a trong ndependent domnatng et of G of Example : Conder the followng graph G mnmum cardnalty Therefore RR 8 But SRR, SRR, SR4R, SR5R, SR6R, SR7R not trong effcent domnatng et of G SR3R the trong effcent domnatng et of G Therefore γre R 9 { vrr, vr5r, vr6r, vr8r, vr9r, vr0r, vr3r, vr5r, vr6r, vrr, vrr, vr3r SR8R SR6 R, R SR7 mnmal trong ndependent domnatng et of maxmum cardnalty Therefore βrr SR8R a mnmal Fgure 8: trong domnatng et of G of maxmum { vrr,vrr,r RvR3R,vR4 R} a trong domnatng et of G cardnalty Therefore ΓRR Therefore ThereforeγRR 4 γr R < RR < γre R < βrr < ΓRR { vrr,vr8r,r RvR9R,vR4 R} an ndependent trong Defnton: 3[4]: A regular pannng ub graph domnatng et of G of degree called -factor (F) Therefore RR 4 Theorem 4: KRn,n F trong effcent and { vrr,vr8r,r RvR9R,vR4 R} a trong effcent domnatng et γrer ( KRn,n F ), R of G ThereforeγRe R 4 Proof: Let G KRn,n F 4 and ΓRR 4 Let V {vrr, vrr,, vrnr, urr, urr,, urnr} ThereforeγRR RR γre R βrr Snce we remove F from KRn,nR, degree of each ΓRR vertex reduced to n Each vr R not adjacent to one urjr, to n and Example j to n The followng example G how that trct Such {vrr, urjr} a trong effcent domnatng et nequalte occur n the above chan Hence γrer ( KRn,n F ), R Example 5: Conder the followng graph KR4,4 F Fgure 9 SR0 R {vrr, vrr, vr3r, vr4r, vr5r, vr6r, vr7r} a trong domnatng et of G of mnmum cardnalty Therefore γrr 7 {vrr, vr3r, vr5r, vr7r, vr5r, vr6r, vr9r, vr0r SRR 8 {vrr, vr3r, vr6r, vr5r, vr6r, vr7r, vrr, vrr, vr3r SRR 9 {vrr, vr4r, vr7r, vrr, vrr, vr3r, vr7r, vr9r, vr0r SR3R 9 Fgure 0 {vrr,urr {vrr,urr {vr3r,ur3r} and {vr4r,ur4r} are trong effcent domnatng et γrer ( KR4,4 F ) Theorem 6: [KRnR] trong effcent and γre n R[KRnR] p ( [KRnR] ) 3n+4 where 75

5 vrr for P P form P P P be can can R R Fgure IJISET - Internatonal Journal of Innovatve Scence, Engneerng & Technology, Vol Iue 4, June 04 p V ( [KRnR] ) Proof: Let n 3 Let vrr, vrr,, vrnr the vertce of KRnR Let G [KRnR] V( [KRnR] ) { vrr, vrr,, vrnr, urr, urr,, u n } By the defnton of [KRnR], each urr adjacent to exactly vertce of KRnR Therefore V ( [KRnR] ) p n + ( n ) n + n(n ) deg vrr n +n any, to n Each vr R adjacent to the remanng (n )vrr and (n )urjr Therefore (n )+ (n ) n Total number of urjr ( n ) n(n ) n n Therefore Number of urjr whch are not adjacent to ( n n ) (n ) n 3n+ Thee n 3n+ urj R together wth vrr a trong effcent domnatng et S of G Therefore G trong effcent S + n 3n+ n 3n+4 Therefore γre R n 3n+4 Let T be any trong effcent domnatng et of [KRnR] Snce T ndependent, T can contan at mot one vr R, n Snce for n 3, no urjr trongly domnate any vrr, T contan at leat one vr R, ( n) Therefore T contan exactly one vr R Any urjr domnate only two vrr and all urjr are ndependent Therefore T contan all urjr Pnot adjacent wth the T therefore T + ( n n ) ( n ) vr R n 3n+4 Therefore γre R([KRnR]) n 3n+4 wwwjetcom Hence γre R([KRnR]) n 3n+4 When n, [KRnR] CR3R γre R(CR3R) Hence γrer[krr] 3()+4 Thu γre R([KRnR]) n 3n+4 Alo p n +n n 3n+4 γre R p ( n ) n +n 4n+4 Example 7: Conder the followng graph G [KR4R] ISSN { vrr, ur3r, ur4r, ur6r { vrr, urr, ur4r, ur5r { vr3r, urr, ur5r, ur6r { vr4r, urr, urr, ur3r are the trong effcent domnatng et of G and p 0, 6 Therefore γre R[KR4R] 4 p Strong effcent domnatng et of dfferent cardnalte n a graph In any graph G admttng effcent domnatng et, all effcent domnatng et have the ame cardnalty namely γ Th not true n the cae of trong effcent domnaton The maxmum cardnalty of any trong effcent domnatng et of G called the upper trong effcent domnaton number of G and denoted by ΓRe R In fact there are graph n whch γre R < ΓRe R and for every potve nteger k wth γre R k ΓRe R, there are mnmal trong effcent domnatng et of dfferent cardnalty k The followng example llutrate th tuaton Let G (V,E) be a mple graph V {vrr, vrr, vrnr, urr, urr,, urn+kr, arr, arr, arn+k+rr, brr, brr,, brn+k+r+r,, wrr, wrr,,wrn+k+r++ +tr, crr, crr,, crkr, drr, drr,,drrr,,err, err,ertr} E vu, au, bu,, wu, n + k va, ua, ba,, wa, n + k + r vb, ub, ab,, wb, n + k + r + vw, aw, bw,, n + k + + t uv, av, bv,, wv, n vc, uc, ac, bc,, wc, k vd, ud, ad, bd,, wd, r vc, uc, ac,, wc, t 76

6 SRR SR3R SRnR SRR { SR4R {wrr, IJISET - Internatonal Journal of Innovatve Scence, Engneerng & Technology, Vol Iue 4, June 04 deg vrr deg urr deg arr Deg wrr (n+k) + (n+k+r) + (n+k+r+) + + (n+k+r+ + +t) Therefore any trong effcent domnatng et of G mut contan only one of the vertce vrr, urr, arr, brr, crr, wrrthe et { vrr, vrr, vrnr } SRR { urr, urr,, urn+kr } { arr, arr, arn+k+rr } { wrr, wrr,,wrn+k+r++ +tr} are trong effcent domnatng et of G SRR n SRR n+k SR3R n+k+r,, SRnR n+k+r+ + +t Therefore γre R n and ΓRe R n+k+r+ + +t Illutraton 8: Let G (V,E) be a mple graph V {vrr, vrr, urr, urr, ur3r, wrr, wrr, wr3r, wr4r, xrr, xrr, xr3r, xr4r, xr5r, arr, brr, crr} E vu, wu, xu 3 vw, uw, xw, 4 vx, ux, w x, 5 av, au, aw, ax, bv, bu deg vrr deg urr deg wrr deg xrr, b w, b x, c v wwwjetcom, c u () (), c w, c x ISSN Neceary and uffcent condton for extence of trong effcent domnatng et Strong effcent graph n whch every vertex contaned n a mnmum trong effcent domnatng et Reference [] DW Bange, AEBarkauka and PJ Slater, Effcent domnatng et n graph, Applcaton of Dcrete Mathematc, 89 99, SIAM, Phladepha, 988 [] DW Bange, AEBarkauka, LHHot and PJ Slater, Generalzed domnaton and effcent domnaton n graph, Dcrete Mathematc,59 (996), [3] JABondy and USR Murty, Graph Theory wth Applcaton, The Macmllan Pre Ltd, 976 [4] FHarary, Graph Theory, Addon Weley, 969 [5] T W Hayne, Stephen T Hedetnem, Peter J Slater Fundamental of domnaton n graph [6] ESampathkumar and L Puhpa Latha Strong weak domnaton and domnaton balance n a graph, Dcrete Math, 6: 35 4, 996 Fgure : { vrr, vrr SRR ur, RuRR, ur3r SR3R wrr, wr3r, wr4r, and { xrr, xrr, xr3r, xr4r, xr5r,}are dfferent trong effcent domnatng et of dfferent cardnalte Therefore γre R and ΓReR 5 Further area of tudy: Characterzaton of trong effcent domnatng et 77