World Journal of Engineering Research and Technology WJERT

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1 wjert 207 Vol. 3 Issue Orgnl Artcle IN X World Journl of ngneerng Reserch nd Technology hndrmouleeswrn et l. World Journl of ngneerng Reserch nd Technology WJRT JIF Impct Fctor: ONNTD WIGHT DOMINATING DG T ON - VALUD GRAPH. Mngl Lvny nd M. hndrmouleeswrn* 2 ree owdmbk ollege of ngneerng Aruppukott - Tmlndu Ind. 2 v Bhnu Kshtry ollege Aruppukott Tmlndu Ind. Artcle Receved on 30/06/207 Artcle Revsed on 2/07/207 Artcle Accepted on /08/207 *orrespondng Author ABTRAT M. hndrmouleeswrn Recently hndrmouleeswrn et.l. ntroduced the noton of v Bhnu Kshtry semrng vlued grphs. nce then severl propertes of -Vlued ollege Aruppukott grphs hve been studed by others. In our erler pper we studed the Tmlndu Ind. noton of edge domnton on -Vlued grphs nd trong nd Wek edge domnton on -Vlued grphs. In ths pper we study the concept of connected weght domntng edge set on -Vlued grphs. KYWORD: dge domnton dge domnton number onnected weght edge domnton.. INTRODUTION The study of domnton set n grph theory ws formlsed s theortcl re n grph theory by Berge. [2] The concept of edge domnton number ws ntroduced by Gupt [3] nd Mtchell nd Hedetnem. [8] mpth Kumr nd Wlkr [0] estblshed the new concept of domnton clled the connected domnton number of grph. The connected edge domnton n grphs ws ntroduced by Arumugm nd Velmml. [] In [9] the uthors hve ntroduced the noton on semrng vlued grphs. In [4] nd [5] the uthors studed the concept of vertex domnton nd connected weght domntng vertex set on -vlued grphs. In [7] we studed the noton of edge domnton on -vlued grphs. Motvted by the work on connected edge domnton on crsp grph [] n ths pper we study the concept of connected weght domntng edge set on -vlued grphs

2 hndrmouleeswrn et l. World Journl of ngneerng Reserch nd Technology 2. PRLIMINARI In ths secton we recll some bsc defntons tht re needed for our work. Defnton 2.: [6] A semrng ( +.) s n lgebrc system wth non-empty set together wth two bnry opertons + nd. such tht () ( + 0) s monod. (2) (. ) s semgroup. (3) For ll b c. (b+c) =. b +. c nd (+b). c =. c + b. c (4) 0. x = x. 0 = 0 x. Defnton 2.2: [6] Let ( +.) be semrng. s sd to be noncl Pre-order f for b b f nd only f there exsts n element c such tht + c = b. Defnton 2.3: [] An edge domntng set X of s clled connected edge domntng set of G f the nduced subgrph X s connected. The connected edge domnton number G (or c for short ) of G s the mnmum crdnlty tken over ll connected edge domntng sets of G. c Defnton 2.4: [9] Let G = (V V V ) be gven grph wth V. For ny semrng ( +.) semrng-vlued grph (or - vlued grph) G s defned to be the grph G V where :V nd : s defned to be mn ( x) ( y) ( x y) 0 otherwse f ( x) ( y) or ( y) ( x) for every unordered pr (x y) of V V. We cll - vertex set nd - edge set of - vlued grph G. Henceforth we cll vlued grph smply s - grph. Defnton 2.5: [4] A vertex u v u N v. v G s sd to be weght domntng vertex f Defnton 2.6: [4] A subset u v u N. v D v D V s sd to be weght domntng vertex set f for ech Defnton 2.7: [7] onsder the -vlued grph G V. An edge be weght domntng edge f e e e N e. eg s sd to 285

3 hndrmouleeswrn et l. World Journl of ngneerng Reserch nd Technology Defnton 2.8: [7] onsder the -vlued grph G V. A subset D s sd to be weght domntng edge set f for ech e D e e e N e. Defnton 2.9: [7] onsder the - vlued grph G V. If D s weght domntng edge set of G then the sclr crdnlty of D s defned by D e. Defnton 2.0: [5] onsder the -vlued grph G V. A connected weght domntng vertex set connected subgrph of G. D V of G s weght domntng vertex set tht nduces Defnton 2.: [9] A -vlued grph G V s sd to be edge regulr -vlued grph f e for ll e nd some. ed 3. onnected weght domntng edge set on -Vlued Grphs In ths secton we ntroduce the noton of onnected weght domntng edge set on -vlued grph nlogous to the noton n crsp grph theory nd prove some smple results. Defnton 3.: onsder the - vlued grph G V. A weght domntng edge set F of s clled connected weght domntng edge set of G f the nduced subgrph F s connected. xmple 3.2: Let 0 b c be semrng wth the followng yley Tbles: + 0 b c 0 0 b c b b b b c c b c. 0 b c b 0 b b b c 0 b b b Let be cnoncl pre-order n gven by b 0 c b b b c c c c b onsder the -grph G V 286

4 hndrmouleeswrn et l. World Journl of ngneerng Reserch nd Technology Defne :V by v v v v v b v c nd : by e e e e e e b e e. lerly e e e e c F s weght domntng edge set nd lso 8 F s connected. Therefore F s connected weght domntng edge set of G. Here F e e e F e e e F e e e F e e e re ll connected weght domntng edge sets. Defnton 3.3: onsder the - vlued grph G V If F s connected weght domntng edge set of G then the sclr crdnlty of F s defned by F e. In exmple 3.2 the sclr crdnlty of F F F F F Defnton 3.4: onsder the - vlued grph G V.A subset F s sd to be mnml connected weght domntng edge set of G f () F s connected weght domntng edge set. (2) No proper subset of F s connected weght domntng edge set. 7 ef In exmple 3.2 F F2 F3 F4 re ll mnml connected weght domntng edge sets. Defnton 3.5: onsder the - vlued grph G V. The connected edge domnton number of G G G denoted by G s defned by G F F s the mnml connected weght domntng edge set. where F In exmple 3.2 F F2 F3 F4 re ll mnml connected weght domntng edge sets wth connected edge domnton number G G F F F F F F F F

5 hndrmouleeswrn et l. World Journl of ngneerng Reserch nd Technology Remrk 3.6: Mnml connected weght domntng edge set n -vlued grph need not be unque n generl. In exmple 3.2 F F2 F3 F4 re ll mnml connected weght domntng edge sets. Remrk 3.7: From the defnton t follows tht ny connected weght domntng edge set s weght domntng edge set n whch the nduced subgrph s connected. Thus we hve every connected weght domntng edge set s weght domntng edge set. However the converse need not be true s seen from the followng exmple. xmple 3.8: onsder the semrng 0 b c exmple 3.2. onsder the -grph G V wth cnoncl pre-order gven n Defne :V by v v v v v b v c nd : by e e b e e e e e Here F e 5 e 7 s weght domntng edge set of G. But Therefore F s not connected weght domntng edge set of G. F s not connected. Remrk 3.9: We observe tht ny subgrph nduced by subset of edges of G my be connected. But the edge set need not be weght domntng edge set s seen n the followng exmple. xmple 3.0: onsder the semrng 0 b c exmple 3.2. onsder the -grph G V wth cnoncl pre-order gven n 288

6 hndrmouleeswrn et l. World Journl of ngneerng Reserch nd Technology Defne :V by v v3 v2 b : by e e b e. 2 3 Here F e e 3 s subset of edges of G s connected but F s not weght domntng edge set. Defnton 3.: onsder the - vlued grph G V.A subset F s sd to nd be mxml connected weght domntng edge set of G f () F s connected weght domntng edge set. (2) If there s no subset F of such tht F F nd F s connected weght domntng edge set. In exmple3.2 F s mxml connected weght domntng edge set. Theorem 3.2: A - vlued grph G wll hve connected weght domntng edge set f nd only f t s connected. Proof Let V be the connected components of G =2...m where \ V \. Let F be the weght domntng edge set of whose elements hs mxml -vlue. nce weght domntng edge set F of G wll hve n edge from every component of G F m F Now F s connected weght domntng edge set F s connected. there exsts common edge e F nd e j F j j nd j=2...m. G s connected

7 hndrmouleeswrn et l. World Journl of ngneerng Reserch nd Technology Theorem 3.3: A - vlued grph G G G 3 G 20 s connected weght domntng edge set then Proof By defn very connected weght domntng edge set s necessrly weght domntng edge set. G G Let F be weght domntng edge set of G such tht G F F let the nduced subgrph F hve m components tht F m. lm There exsts nd shortest pth between j where j be two components of F such tht the length of nd j s tmost 3 n G. Assume tht there exst shortest pth between nd j of length tlest 4. Let P be the shortest pth between the components of nduced subgrph F. Tht s P s the shortest of ll the shortest pth between ny two dstnct components of F. Hence we cn fnd n edge from the end ponts of P. e e n the pth P such tht e s t dstnce of tlest 2 nce F s weght domntng edge set then the edge e must be t dstnce of tmost from component. Thus the edge e les on pth P between the two components such tht P s shorter thn P. Ths contrdcts the ssumpton tht the length of the pth P s tlest

8 hndrmouleeswrn et l. World Journl of ngneerng Reserch nd Technology Ths proves tht there exsts two components nd j where j of F such tht the pth between the two hs tmost 3. Addng edge n the pth P to the weght domntng edge set F decreses the number of components of F by. ontnung ths procedure we obtn only one component n weght domntng edge set. F provng tht $F$ s Thus we cn dd tmost 2(m-) edges to the weght domntng edge set F so s to form connected weght domntng edge set. Thus m G F F 2 e m F F 2 e 0 F ef F F 2F F 0 G 2 G 0 G onnected edge domnton number for omplete edge regulr grphs In ths secton we study through some exmples how to fnd connected edge domnton number G for gven complete edge regulr grph G. For ny complete edge regulr - vlued grph G on n vertces wth weght for ll edges. G n 2 Let G G2 G3 be three complete edge regulr grphs wth 34 nd 5 vertces wth weght for ll edges respectvely. 29

9 hndrmouleeswrn et l. World Journl of ngneerng Reserch nd Technology ( 2 3 Here G ) G ( 2) G ( 3) Let G G 4 5 be two complete edge regulr grphs wth 6 nd 7 vertces wth weght for ll edges respectvely. 4 ( 5 Here G 4) G ( 5) From the bove study of exmples we cn obtn the followng lgorthm for complete edge regulr grph on $n$ vertces wth weght for ll edges respectvely. n () onsder complete edge regulr grph K. (2) Frst fnd n rbtrry edge $e \n $ of the complete edge regulr K. (3) Fnd N e. (4) Tke ny one of the edge e N e (5) Now fnd N e.. (6) Then tke ny one of the edge e N. 2 e n 292

10 hndrmouleeswrn et l. World Journl of ngneerng Reserch nd Technology (7) ontnung ths the process wll termnte fter fnte number of steps (.e f the collecton of such edges domntes ll the edges of the complete grph K. (8) Ths collecton of edges form mnmum connected weght domntng set for complete n grph K. n RFRN. Arumugm. nd Velmml. : dge domnton n Grphs Twnese Journl of Mthemtcs 998; 2(2): Berge : Theory of Grphs nd ts Applctons Methuen London Gupt. R. P: Independence nd overng numbers of the lne grphs nd totl grphs Proof technques n grph theory (ed. F.Hrry) Acdemc press New York 969; Jeylkshm. nd hndrmouleeswrn.m: Vertex domnton on - vlued Grphs IOR Journl of Mthemtcs (IOR-JM) 206; 2(III): Jeylkshm. nd hndrmouleeswrn.m: onnected weght domntng vertex set on - vlued Grphs IJPAM 207; 2(5): Jonthn Goln emrngs nd Ther Applctons Kluwer Acdemc Publshers London. 7. Kruthg Deep. Mngl Lvny. nd chndrmouleeswrn.m: dge Domnton on - vlued grph Journl of mthemtcl nd computtonl cence 207; 7(): IN: JMtchell. nd Hedetnem: dge domnton n trees ongr. Numer 977; 9: Rjkumr.M. Jeylkshm. nd hndrmouleeswrn.m: emrng-vlued Grphs Interntonl Journl of Mth. c. nd ngg. Appls. 205; 9(III): mpth Kumr. nd Wlkr.H.B: onnected domnton number of grph J.Mth.Phy.c. 979; 3(6): Velmml: qulty of onnected dge Domnton nd Totl dge Domnton n Grphs Interntonl Journl of nhnced Reserch n cence Technology nd ngneerng IN: My-204; 3(5):