Different types of Domination in Intuitionistic Fuzzy Graph

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1 Aals of Pur ad Appld Mathmatcs Vol, No, 07, 87-0 ISSN: X P, ol Publshd o July 07 wwwrsarchmathscorg DOI: Aals of Dffrt typs of Domato Itutostc Fuzzy Graph MGaruambga, SSasakar ad Palal 3 Dpartmt of Mathmatcs, Sr Vasa Collg Erod , Tamladu, Ida -mal:karusc@yahoo Dpartmt of Scc ad Humats, PES Ursty Bagalor , arataka, Ida 3 Dpartmt of Mathmatcs, Th Oxford Collg of Egrg Bagalor , arataka, Ida -mal: skars@gmalcom Corrspodg author -mal: sshakar@gmalcom Rcd Ju 07; accptd July 07 Abstract I ths papr, w troducd a dfto of dg domato st usg strog dgs ad dg dpdt sts of tutostc fuzzy graphs W dtrm th domato umbr γ G ad dg domato umbr γ G for sral classs of tutostc fuzzy graphs ad th rlato btw thm ar dscussd Also w troduc a rgular domatg st ad rgular dpdt st tutostc fuzzy graphs wth sutabl llustratos ywords: Edg domatg st IFG, dg domato umbr, dg dpdt st, rgular domato st AMS Mathmatcs Subct Classfcato 00: 05C7, 03E7, 03F55 Itroducto Th study of domatg sts graph was troducd by Or ad Brg 96 Th dg domato was troducd by Arumugam ad Vlammal [] Th dg domato problm has may applcatos rsourc allocato, twork routg ad codg thory problms [, 3] Somasudaram ad Somasudaram [7] troducd th cocpt of domato fuzzy graphs ad obta sral bouds for th domato umbr Domato fuzzy graphs usg strog dgs was dscussd by Nagoorga ad Chadraskara [8] Parath ad Thamzhdh [] troducd domatg st, domato umbr, dpdt st, total domatg ad total domato umbr tutostc fuzzy graphs Rsarch work of sral stgators [, 5, 6, 9, 3, 5, 6] ha motatd us to dlop th dffrt typs of domato tutostc fuzzy graphs Ths papr s orgazd as follows Scto cotas prlmars ad scto 3, a dg domato umbr ad dg dpdt umbr of a IFG s dfd usg 87

2 MGaruambga, SSasakar ad Palal strog dgs Scto dals wth proprts of dg domato IFGs ad th rlato btw domato umbr γ G ad dg domato umbr γ G Fally, w troducd rgular domatg st ad rgular dpdt st IFG scto 5 Prlmars Dfto [0] A tutostc fuzzy graph IFG s of th form 88 G V, E sad to b a mmax IFG f V {, } such that : V [0,] ad ν : V [0,], dot th dgr of mmbrshp ad o-mmbrshp of th lmt 0 + ν, for ry V,,,,, V rspctly ad E V V whr : V V [0,] ad ν : V V [0,], ar such that m[, ], ν max[ ν, ν ], dots th dgr of mmbrshp ad o-mmbrshp of th dg rspctly, whr + ν, for ry E 0 E For ach tutostc fuzzy graph G, th dgr of hstachstato dgr of th rtx V G s π ν ad th dgr of hstac hstato dgr of a dg E G s π ν Dfto [] A IFG, G V, E s sad to b complt IFG I m, ad ν max ν, ν for ry V Dfto 3 [] A IFG, G V, E s sad to b strog IFG f m, ad ν max ν, ν for ry E Dfto [] Th complmt of a IFG, G V, E s a IFG, G V, E, whr V V, ad ν ν, for all,,,, 3 m, ad ν max ν, ν ν for all,,,, Dfto 5 [7] Th ghbourhood dgr of a rtx s dfd as d d, d ν whr N N N d N ν w N w N w ad d N ν w

3 Dffrt typs of Domato Itutostc Fuzzy Graph Dfto 6 [8] Lt G V, E b a IFG Th th dgr of a rtx s dfd by d d, d, G ν whr d ad dν ν Dfto 7 [8] A tutostc fuzzy graph G V, E s sad to b a, -rgular f d, G for all V ad also G s sad to b a rgular tutostc fuzzy graph of dgr, Dfto 8 [] A tutostc fuzzy graph G V, E s sad to b a bpartt f th rtx st V ca b parttod to two o mpty sts V ad V such that 0 ad ν 0 f V or V > 0, ν < 0 f V or V, for som ad, or 0, ν < 0 f V or V, for som ad, or > 0, ν 0 f V or V, for som ad Dfto 9 [] A bpartt tutostc fuzzy graph G V, E s sad to b complt f m, ad ν max ν, ν for all V ad V ad s dotd by Dfto 0 [] If ad btw V, V V G, th -strgth of coctdss btw k s sup{ k,, } ad ν -strgth of coctdss ad k s ν f{ ν k,, } k If u ar coctd by mas of paths of lgth k th u, s dfd as sup u, u, } ad { 3 k k V k u, f { ν u, ν, ν, 3 ν k, u,, k, V ν s dfd as Dfto [] A dg u s sad to b a strog dg f u, u, ad ν u, ν u, } Dfto [] Lt G V, E b a IFG o V Lt u V, w say that u domats G f thr xsts a strog dg btw thm Dfto 3 [] A subst S of V s calld domatg st G f for ry V S, thr xsts u S such that u domats Dfto [] A domatg st S of a IFG s sad to b mmal domatg st f o propr subst of S s a domatg st 89

4 MGaruambga, SSasakar ad Palal Dfto 5 [] Mmum cardalty amog all mmal domatg st s calld rtx domato umbr of G ad s dotd by γ G Dfto 6 [] Lt G V, E b a IFG, th th rtx cardalty of V s dfd by + γ V for all V V Dfto 7 [] Lt G V, E b a IFG, th th dg cardalty of E s + γ dfd by E for all E E 3 Edg domato tutostc fuzzy graphs Th cocpt of dg domato graphs was troducd by Arumugam ad Vlammal 988 ad furthr dg domato ad dpdc fuzzy graphs s studd by Nagoorga ad Prasaa d W rfr to [] for th trmology of domato tutostc fuzzy graphs Dfto 3 Lt G V, E b a IFG Lt ad W say that domats f s a strog dg G b two adact dgs of G Dfto 3 A subst S of E s calld a dg domatg st G f for ry E S, thr xsts S such that domats Dfto 33 A dg domatg st S of a IFG s sad to b mmal dg domatg st f o propr subst of S s a dg domatg st Dfto 3 Mmum cardalty amog all mmal dg domatg st s calld dg domato umbr of G ad s dotd by γ G Dfto 35 Th strog ghbourhood of a dg a IFG G s N { E G s strog dg ad adact to G } s Exampl 3 Cosdr a IFG, G V, E, such that V {, } ad E {,,, 3, 3,, 3, 5,, 5,, 6, 6, } Fgur : 90

5 Dffrt typs of Domato Itutostc Fuzzy Graph Hr, },,, },, },,, },, } ar mmal dg domatg { { 5 sts of G ad γ G 07 { 5 { 7 { 6 7 Dfto 36 Lt E b a subst of dg st E Th th od cor of as th st of all rtcs cdt to ach dg E E s dfd Dfto 37 Two dgs ad G V, E f N ad s ar sad to b dg dpdt a IFG N s Dfto 38 A subst S of E s sad to b a tutostc fuzzy dg dpdt st of a IFG G f ay two dgs S ar dg dpdt Dfto 39 A dg dpdt st S of G a IFG s sad to b maxmal dpdt f for ry dg E S, th st S {} s ot dpdt Dfto 30 Th mmum cardalty amog all maxmal dg dpdt st s calld dg dpdc umbr of G ad t s dotd by β G Dfto 3 A dg IFG G s calld a solatd dg f t s ot adact to ay strog dg G Exampl 3 Cosdr a IFG, G V, E, such that V {, } ad E {,,, 3, 3,,,, 5,,, 5, } 3 5 Fgur : Hr, },, },, },, },, },, },, } ar maxmal { 3 { { { 5 9 { 3 5 dg dpdt sts of G ad β G 075 { 5 7 { 6 Dfto 3 If all th dgs ar strog dg a IFG th t s calld strgthd IFG Thorm 3 Nod cor of a dg domatg st of a IFG G s a domatg st of G Proof: Lt V b th od cor of a dg domatg st D of G W ha to pro that V s a domatg st of G Suppos V s ot a domatg st Th thr xsts atlast o rtx u whch

6 s ot domatg by th lmts of MGaruambga, SSasakar ad Palal V V dos ot cota ay rtx from N s u u for all Th D dos ot cota ay strog dg whch domats N s u Hc D s ot a dg domatg st of G Ths s a cotradcto Thrfor od cor of a dg domatg st of a IFG s a domatg st of G Thorm 3 Lt G b a IFG wthout solatd dgs ad thr xsts o dg E such that N s D If D s a mmal dg domatg st, th S D s a dg domatg st, whr S s th st of all strog dg G Proof: Lt D b a mmal dg domatg st of a IFG G Suppos S D s ot a dg domatg st Th thr xsts atlast o dg D whch s ot domatd by S D Sc G has o solatd dgs ad thr s o dg E such that N s D, s adact to atlast o strog dg dg domatg st of G, S D Hc D 9 S Sc S D s ot a Thrfor D { } s a dg domatg st whch s cotradcto to th fact that D s mmal dg domatg st Thus, ry dg E S s domatd by a dg S D Thrfor S D s a dg domatg st Thorm 33 A dg dpdt st of a IFG hag oly strog dgs s a maxmal dg dpdt st f ad oly f t s dg dpdt ad dg domatg st Proof: Lt S b a dg dpdt st of a IFG hag oly strog dgs Suppos S s a maxmal dg dpdt st of G Th for ry E S, th st S } s ot a dg dpdt st, for ry E S, thr s a dg { such that N Thus, S s a dg domatg st of G ad also t s a dg s dpdt st ofg Corsly, Suppos S s both dg dpdt ad dg domatg st of G W ha to pro that S s maxmal dg dpdt st hag oly strog dgs Sc S s a dg domatg st of G, t has oly strog dgs Assum that S s ot a maxmal dg dpdt st Th thr xsts a dg S such that S { } s a dg dpdt st, thr s o dg S blogg to N ad thrfor s ot domatd by S Hc S caot b a dg domatg st of G, whch s a cotradcto Hc S s a maxmal dg dpdt st ofg hag oly strog dgs Thorm 3 Ery maxmal dg dpdt st a IFG G hag oly strog dgs s a mmal dg domatg st of G Proof: Lt S b a maxmal dg dpdt st hag oly strog dgs of a IFG G s

7 Dffrt typs of Domato Itutostc Fuzzy Graph By Thorm 33, S s a dg domatg st of G Suppos S s ot a mmal dg domatg st of G Th thr xsts a dg S such that S { } s a dg domatg st Th atlast o dg S { } s N Ths cotradcts th fact that S s a dg dpdt st of G s Hc S s a mmal dg domatg st of G Thorm 35 Nod cor of a maxmal dg dpdt st of a IFG hag oly strog dgs s a domatg st of G Proof: Lt S b a maxmal dg dpdt st of a IFG G hag oly strog dgs Lt V b th od cor of S By Thorm 3, Ery maxmal dg dpdt st hag oly strog dgs s a mmal domatg st of G Th V s a od cor of a dg domatg st of a IFG G By Thorm 3, od cor of a dg domatg st of a IFG G s domatg st of G Hc V s a domatg st of G Thorm 36 Ery complt IFG s strgthd IFG Proof: Lt G V, E b a complt IFG By dfto of complt IFG u m u, ad ν u, max ν u, ν, for all u, V Suppos G has atlast o o strog u dg th u < u, ad ν u < ν u, whch mpls that u < m u, ad ν u, < max ν u, ν, for som u V Ths cotradcts our assumpto that G s complt IFG Thus, ry dg complt IFG s a strog dg Hc ry complt IFG s strgthd IFG Corollary 37 Ery strog IFG s a strgthd IFG Corollary 38 Ery slf complmtary IFG s a strgthd IFG Corollary 39 Ery complt bpartt IFG, s a strgthd IFG Thorm 30 Lt G V, E s a IFG ad f G s both rgular ad dg rgular IFG th, s a costat fucto ν Proof: Assum that G s both rgular ad dg rgular IFG By dfto of rgular IFG, d d, d, G ν V By dg rgular IFG, dg d, dν l, l E whr d d + d E l + 93

8 MGaruambga, SSasakar ad Palal l E ad d d + d ν E ν ν ν l + ν l ν E Hc, s a costat fucto ν Not Lt G V, E b a complt IFG wth, ν as a costat fucto ad f G s, rgular IFG th ad ν E Thorm 3 Lt G V, E b a slf complmtary IFG Th G s a, dg rgular IFG f ad oly f G s also, dg rgular IFG Proof: Sc G V, E s a slf complmtary IFG m, / ad ν max ν, ν / V By th dfto of complmt, m, ad ν max ν, ν ν V Thrfor m, / ad ν max ν, ν / V Hc ad ν ν Now d d, d G ν E k k E k + k k k d k k E k k E + k k d d Smlarly d d ν ν Thrfor d d E G G Hc G s, dg rgular IFG f ad oly f G s also, dg rgular IFG 9

9 Dffrt typs of Domato Itutostc Fuzzy Graph Rlato btw domato umbr γ G ad dg domato umbr γ G of a IFG Notato Lt P b th umbr of dgs mmum dg domatg st of Lmma P, wh s Proof: It ca b asly rfd that P ad P Wh 6, cosdr 6 Lt E 6 {,, 5} ad f P 6 th N ad N costtut 9 dgs ad sc E 6 5 th rmag 6 dgs ar ducd graph of G [ 3,, 5, 6] whch s clarly But P P 6 + P + + P tms Wh 8, cosdr 8 Lt E 8 {,, 8} ad f P 8 th N ad N costtut 3 dgs ad sc E 8 8 th rmag 5 dgs ar ducd graph of G [ 3,, 5, 6, 7, 8 ] whch s clarly 6 But P 6 3 P + P + + P tms 8 6 Smlarly, cosdr E {,, Lt } ad f P th N ad N costtut 3 dgs ad sc E th rmag dgs ar ducd graph of G, ] whch s [ clarly P + P [+ P [+ + P ] + [+ + + tms] + 6 P + Hc P, wh s Lmma P, wh s odd Proof: It ca b asly rfd that P 0 ad P 3 Wh 5, cosdr 5 Lt E 5 {,, 0} ad f P 5 th N ad N costtut 7 dgs ad sc E 5 0 th rmag 3 dgs ar ducd graph of G [ 3,, 5] whch s clarly 3 But P 3 P 5 + P P + tms Wh 7, cosdr 7 Lt E 7 {,, } ad f P 7 th N ad N costtut dgs ad sc E 7 th rmag 0 ] 95

10 MGaruambga, SSasakar ad Palal dgs ar ducd graph of G [ 3,, 5, 6, 7 ] whch s clarly 5 But P 5 P + P + + P tms Smlarly, cosdr Lt E {,, } ad f P th N ad N costtut 5 dgs ad sc E 3 th rmag dgs ar ducd graph of + G [,,, 3] whch s clarly 3 P P [+ + 7 P Hc P ] + [+ + + tms] P, wh s odd Lmma 3 If G V, E s a complt IFG th P m P m + P Proof: Cas Wh m ad s m + m P m+ + P m + P Cas Wh m s odd ad s m + m P m + P m + P + Rmark Th abo rsult s ot tru whr m ad s both odd + + Lmma For ay complt IFG, P + Proof: To pro P, w us th mthod of ducto Cas wh s odd For, P 0 s a sgl rtx Assum that th rsult s tru for all m +, w ha to pro for m + 3 m+ m + + P m + m P m + 3 P m+ + P By Lmma 3 P m + + m + Hc P m+ 3 s tru Cas wh s

11 Dffrt typs of Domato Itutostc Fuzzy Graph For, P s a dg Assum that th rsult s tru for m, w ha to pro for m + m m + P m m P m + P m + P By Lmma 3 m m + + P + + m + Hc P m+ s tru Hc th proof m + Lmma 5 If P dot th umbr of dgs mmal dg domatg st of complt graph th P + P, wh s P + P +, wh s odd Proof: Wh s P + P Hc P + P, wh s Wh s odd + P P + Hc P P +, wh s odd Thorm 6 For a complt IFG G wth rtcs, s, s odd whr γ G ad γ G s th mmum cardalty of th rtx ad dg domatg st +

12 MGaruambga, SSasakar ad Palal Proof: Lt G V, E b a complt IFG wth By Thorm 36, G s strgthd IFG If w choos a dg u th t wll domat all th dgs cdt to u ad Th st of dgs u whr o two of thm has a rtx commo forms a mmal dg domatg st Mmum cardalty of dg domatg st s γ G Sc G s complt IFG, mmum cardalty of rtx domatg st wll b oly o rtx ad t s γ G If s, th P cotas dgs By Lmma γ G s th sum of mmum cardalty of dgs ad γ G s a mmum cardalty of a rtx Sc G s complt IFG, th dgs assocatd wth γ G wll ha mmum cardalty Hc th sum of mmum cardalty of dg domatg st γ G γ G Smlarly, f s odd, th P cotas dgs By Lmma ad γ G γ G Hc th rsult Thorm 7 If G V, E s a complt IFG th γ G < γ G f > 3 Proof: Assum that G V, E s a complt IFG Th ry dg G s strog dg γ G s th mmum cardalty of a rtx + By Lmma, P cotas dgs + γ G s th sum of mmum cardalty of If > 3 th γ G cotas mor tha o dg Hc γ G < γ G dgs Not Lt G V, E s a complt IFG wth, as costat fucto ad 3 th γ G γ G ν Not 3 If G s dg rgular IFG wth, ν as costat fucto th G s dg rgular IFG Rmark If γ G ad γ G s domato umbr ad dg domato umbr of a IFG wth > 3 th 98

13 Dffrt typs of Domato Itutostc Fuzzy Graph γ G γ G < γ G, s γ G γ G < γ G, s odd Thorm 8 Lt G V, E b a slf complmtary IFG th γ G γ G Proof: Sc G s slf complmtary IFG Ery dg slf complmtary IFG s a strog dg ad ad ν ν Also G s somorphc to G Thrfor mmum cardalty of dg domatg st of G ad G rmas sam Hc γ G γ G Thorm 9 Lt G V, E b a complt bpartt IFG, ad f G s both rgular ad dg rgular IFG th γ G γ G Proof: G G V, E b a complt bpartt IFG, th ry dg G s a strog dg Assum that G s both rgular ad dg rgular IFG By Thorm 30,, s a costat fucto th th rtx domato umbr γ G u, ν ad th dg domato umbr γ G u, Hc γ G γ G 5 Rgular domato tutostc fuzzy graphs Dfto 5 Lt G V, E b a IFG A st S subst of V s calld rgular tutostc fuzzy domatg st f ry rtx V S s adact to som rtx S all th rtcs S has th sam dgr Exampl 5 Cosdr a IFG, G V, E, such that V {, } ad E {,,, 3, 3, 6,,, 5, 5, 6} Fgur 3: 99

14 MGaruambga, SSasakar ad Palal Dfto 5 Th mmum cardalty of rgular tutostc fuzzy domatg st s calld rgular tutostc fuzzy domatg umbr ad dotd by γ G 08 Dfto 53 A st S subst of V s calld mmal rgular tutostc fuzzy domatg st f Ay subst of S s ot a tutostc fuzzy domatg st all th rtcs of S ha th sam dgr Dfto 5 A dpdt st S of a IFG G V, E s sad to b rgular dpdt IFG f all th rtcs of S has th sam dgr Dfto 55 A st S s maxmal rgular dpdt st f for ry rtx th st S {} s ot rgular dpdt st Exampl 5 Cosdr a IFG, G V, E, such that V {, } ad E {,, 3, 3,,, } rf 3 V S Fgur : Hr, }, } s a rgular dpdt st { 3 { Thorm 5 A rgular dpdt st s a rgular maxmal dpdt st of a IFG f ad oly f t s rgular dpdt ad rgular domatg st Proof: Lt S b a rgular maxmal dpdt st a IFG, th for ry u V S, th st S {u} s ot a dpdt st, for ry u V S, thr s a rtx S such that u s adact to Thus, S s domatg st of G ad also rgular dpdt st of G Hc S s rgular dpdt ad rgular domatg st Corsly, Suppos S s both rgular dpdt ad rgular domatg st of G W ha to pro that S s rgular mxmal dpdt st Assum that S s ot a maxmal dpdt st Th thr xsts a rtx u S such that S {u} s a dpdt st, thr s o rtx S adact to u ad thrfor u s ot domatd by S Hc S caot b a domatg st of G, whch s cotradcto Hc S s maxmal dpdt Thus S s rgular maxmal dpdt st Thorm 5 Ery rgular maxmal dpdt st a IFG G s a rgular mmal domatg st of G Proof: Lt S b a rgular maxmal dpdt st a IFG By Thorm 5, S s 00

15 Dffrt typs of Domato Itutostc Fuzzy Graph rgular domatg st of G Suppos S s ot a mmal domatg st of G Th thr xsts atlast o rtx S such that S {} s domatg st Th atlast o rtx S {} s adact to Ths cotradcts th fact that S s rgular dpdt st of G Hc S s rgular mmal domatg st of G REFERENCES SArumugam ad SVlammal, Edg domato graphs, Tawas Joural of Mathmatcs, Ataasso, Itutostc fuzzy sts: Thory ad applcatos, studs fuzzss ad soft computg, Hdlbrg, Nw York, Physca-Vrl, GJChag, Algorthmc aspcts of domato graphs, : DZ Du, PM Pardalos Eds, Hadbook of Combatoral Optmzato, Vol 3, luwr, Bosto, MA, 998, MGaruambga, SSasakar ad Palal, Som proprts of a rgular tutostc fuzzy graph, Itratoal Joural of Mathmatcs ad Computato, MGaruambga, Palal ad SSasakar, Edg rgular tutostc fuzzy graph, Adacs Fuzzy Sts ad Systms, OTMausha ad MSSutha, Strog domato fuzzy graphs, Fuzzy Iformato ad Egrg, ANagoorga ad S Shatha Bgum, Dgr, ordr ad sz tutostc fuzzy graphs, Itratoal Joural of Algorthms, Computg ad Mathmatcs, ANagoorga ad VTChadraskara, Domato fuzzy graph, Adacs Fuzzy Sts ad Systm, ANagoorga ad Prasaa D, Edg domato ad dpdc fuzzy graphs, Adacs Fuzzy Sts ad Systms, RParath ad MGaruambga, Itutostc fuzzy graphs, Computatoal Itllgc, Thory ad applcatos, RParath, MGaruambga ad Ataasso, Opratos o tutostc fuzzy graphs, Procdgs of IEEE Itratoal Cofrc Fuzzy Systms FUZZ-IEEE, RParath ad GThamzhdh, Domato tutostc fuzzy graphs, Nots o Itutostc Fuzzy Sts, Radha ad Numaral, O dg rgular fuzzy graphs, Itratoal Joural of Mathmatcal Arch, ARosfld, Fuzzy graphs, I Fuzzy Sts ad Thr Applcatos to Cogt ad Dcso Procsss, Acadmc Prss, SSahoo ad MPal, Itutostc fuzzy comptto graphs, Joural of Appld Mathmatcs ad Computg, SSahoo ad MPal, Product of tutostc fuzzy graphs ad dgr, Joural of Itllgt ad Fuzzy Systms, ASomasudaram ad SSomasudaram, Domato fuzzy graphs-i, Pattr Rcogto Lttrs,