Complementary Tree Paired Domination in Graphs

Transcription

1 IOSR Joural of Mahemacs (IOSR-JM) e-issn: , p-issn: X Volume 2, Issue 6 Ver II (Nov - Dec206), PP 26-3 wwwosrjouralsorg Complemeary Tree Pared Domao Graphs A Meeaksh, J Baskar Babujee 2 (Deparme of Mahemacs, SKR Egeerg College, Chea-60023, Ida) 2 (Deparme of Mahemacs, Aa Uversy, MIT Campus, Chea ,Ida) Absrac: A domag se S V s sad o be a complemeary ree domag se f he duced sub graph V S s a ree The mmum cardaly of a complemeary ree domag se of G s called he complemeary ree domao umber of G ad s deoed by DOI: 09790/ wwwosrjouralsorg 26 Page cd I hs paper we have roduced he ew ype of domao amed complemeary ree pared domao umber for coeced graphs ad we sudy he heorecal operes of hs parameer ad may bouds are obaed erms of order of G ad s relaoshp wh oher domao parameers are also obaed The relao bewee he complemeary ree pared domao umber ad oal domao umber of a ree s also dscussed Keywords: Graph, Domao, ree, Perfec Machg I Iroduco Le G( p, q) be a smple udreced graph wh p verces ad q edges The se of verces s deoed by V ; he se of edges by E Order of G s deoed as G as a symbol for he cardaly of V The degree, Neghborhood ad closed eghborhood of a verex v he graph G are deoed by (v) N( v) ad N v N( v) v respecvely For a subse S of V, N (S) deoes he se of all d, verces adjace o some verex of S G ad NS N S) S ( Le S ad G S deoe he sub graphs of G duced by S ad V S respecvely The mmum degree ad maxmum degree of he graph G are deoed by ad ( G) respecvely A verex of degree oe s called a leaf ad s eghbor s a suppor verex A double sar s a ree wh exacly wo suppor verces The coroa of a graph G s he graph formed from a copy of G by aachg for each v V, a ew verex v ' ad edge vv ' I geeral, he K -coroa of a graph G s he graph of order k V obaed from G by addg a pah of legh k o each verex of G so ha he resulg pahs are verex dsjo A se S V s a domag se f for every verex v V S, here exss a verex u S such ha v s adjace o u The mmum cardaly of a domag se G s called he domao umber of G ad s deoed by We call a domag se of mmum cardaly a -se of G A se S V s a oal domag se of abbrevaed TDS, f every verex V s adjace o a verex S We call a oal domag se of mmum cardaly a -se of G A se S V s a pared domag se f he duced sub graph S has a perfec machg The mmum cardaly of a pared domag se G s called he pared domao umber of G ad s deoed by We call a pared domag se of mmum cardaly a -se of G Ths cocep was roduced by Hayes e al [] A domag se D of a coeced graph G s a o spl domag se, f he duced sub graph V D s coeced The mmum cardaly of a o spl domag se of a graph G s called he o spl domao umber ad s deoed by he A domag se D of a coeced graph G s a spl domag se, f he duced sub graph s V D s dscoeced The mmum cardaly of a spl domag se of a graph G s called he spl domao umber s deoed by he Boh hese coceps were roduced by Kull e al[4] A domag se s S V s sad o be a complemeary ree domag se f he duced sub graph V S s a ree The mmum cardaly of a complemeary ree domag se of G s called he complemeary ree

2 Complemeary Tree Pared Domao Graphs domao umber of G ad s deoed by cd The complemeary ree domag se was roduced by SMuhamma e al [7] If G ( p, q) ad G2 ( p2, q2 ) are wo coeced graphs, G oˆg 2 s obaed by mergg ay seleced verex of G 2 o ay seleced verex of G If we arbrarly choose he verces of G ad G 2 he G oˆg 2 s a class of graphs cossg of p p2 verces ad q q2 edges I geeral, we cosruc p p 2 possble combao of graphs from G ad G2 Ths cocep was roduced by Jayapal Baskar Babujee e al [3] The cocep of compleme of a graph was defed by Body e al [] May of he auhors roduced he cocep relaed wh complemes I parcular, complemeary perfec domag se was roduced by Paulraj Josephe al[9]the cocep Complemeary l domao umber of a graph was roduced by TTamzhchelvam e al[0] Lkewse, I hs paper we have roduced he ew parameer amed complemeary ree pared domao umber ad s deoed by We oba he upper bouds ad lower bouds of complemeary ree pared domao umber erms of maxmum degree,sze ad order of he graph G II Complemeary Tree Pared Domao Number Ad Is Lower Ad Upper Bouds Defo 2: A pared domag se S V s sad o be a complemeary ree pared domag se f he duced sub graph V S s a ree The mmum cardaly of a complemeary ree pared domag se (CTPD se) of G s called he complemeary ree pared domao umber of G ad s deoed by ad he correspodg se s se of G The exsece of such a se s guaraeed oly for coeced graphs Example: Le G be a graph meoed fgure () Observao 22: Fg() 2 Some observaos are made 2 f 0(mod 2) For he pah graph P, ( P ) f (mod 2) 2 2 f 0(mod 2) For he cycle graph C, ( C ) f (mod 2) 2 f 0(mod 2) For he complee graph K, ( K ) f (mod 2) 3 For ay coeced graph G, 4 For ay coeced graph G, 2 f 0(mod 2) f (mod 2) The equaly holds good for C ad K DOI: 09790/ wwwosrjouralsorg 27 Page

3 Complemeary Tree Pared Domao Graphs Observao 23[6]: A pared domag se(pds) S of a graph G s a mmal PDS f ad oly f ay wo verces x, y S sasfy oe of he followg codos: () G[ S { x, y}] does o coa a perfec machg () Whou loss of geeraly, x s a ed verex G[ S] adjace o y () There exss a verex u V S suchha N( u) S { x, y} Observao 24: A CTPD se S of a graph G s a mmal CTPD se f ad oly f ay wo verces x, y S sasfy oe of he followg codos: () G[ S { x, y}] does o coa a perfec machg () There exss a verex u V S suchha N( u) S { x, y} Proof: Suppose S s a mmal complemeary ree pared domag se of G The for ay wo verces x ad y, S x, y s o a complemeary ree pared domag se of G Therefore G[ S { x, y}] does o coa a perfec machg Le u V S s o domaed by S x, y, bu s domaed by S he u s adjace o eher xor y or boh Coversely, suppose S s a mmal complemeary ree pared domag se of G The for ay wo verces x, y S, oe of he wo saed codos holds Now ove ha S s a mmal complemeary ree pared domag se of G Suppose S s o a mmal complemeary ree pared domag se,he here exss wo verces say x, y S such ha S x, y s a complemeary ree pared domag se Therefore codo () does o hold If S x, y s a complemeary ree pared domag se he every verex V S s adjace o a leas oe verex S x, y Therefore for ay wo verces x, y S, codo () does o hold Hece eher codo () or () holds, whch s a coradco Nex le us see some lower bouds of CTPD umber Theorem 25: For ay coeced graph, Proof: Le G be ay coeced graph Ay complemeary ree pared domag se of G s also a pared domag se of G Thus Furhermore, ay pared domag se of G s also a domag se of G Hece Therefore we coclude ha HBWalkar e al[2] already oved ha p For ay graph G, p Theorem 26: For ay coeced graph G ( p, q) wh p 2, ( ) 2 - G q p p Proof: Sce ad By heorem 25, we have p () Sce V D s a ree, p Hece p 2( p ) ( p ) 2q p () p From () ad () we have ( ) 2 - G q p Theorem 27: If ( p, q) Proof: Le D be a G s a coeced graph wh 2 he 3p 2q 2 se of G Le be he umber of edges G havg oe DOI: 09790/ wwwosrjouralsorg 28 Page

4 verex D ad oher verex V D The umber of verces ( p ), We kow ha 2 q vd d( v ) vv D Complemeary Tree Pared Domao Graphs V D s V D = p ad he umber of edges V D s d( v ) Sce d( v ) 2[( p ) ] vd vd Sce vv D d( v ) 2q d( v ) vv D d( v ) 2 q (( p ) ) V D p here are a leas p edges from V D o D Also deg ( ) v Hece, 2q ( p ) p Sce 2, we have 2 q p G) 2 p ( G ) 2q 2p p Hece 3p 2q 2 Theorem 28: If ( p, q), 2 G s a coeced graph, he here exss a graph G ', ( G') 2 G o C such ha ( G') 3p 2q 4 k, wherek (mod 2) G ( p, q be a coeced graph wh 2 ad C be a cycle graph G0 C ad le G' G0C V( G0ˆ C ) p ad E( G0ˆ C ) q he class Proof: Le ) Form he graph By heorem 27, 3p 2q 2 ad ( ) f s eve Hece ( G0C ) (3p 2q 2) ( 2) Therefore ( G ') 3p 2q 4 Also ( C ) - f s odd Therefore ( G0C ) (3p 2q 2) ( ) C 2 3p 2q 3 Hece ( G ') 3p 2q 4 k, wherek (mod 2) Theorem 29: If G p, ) ; ( G ) 2 ( q ad G2 ( p2, q2 ) ; ( G2 ) 2 G o such ha 2p q 4 exss a graph G from he class 2 Proof: Le G G 0G2 Le G p, ) be a coeced graph ad G p, ) be a coeced graph ( q G 2 ( 2 q2 from are coeced graphs, he here Form he graph G0G 2 wh p verces ad q edges, where p p p2 ad q q q2 Le D be a se of G Le be he umber of edges G havg oe verex D ad oher verex V D DOI: 09790/ wwwosrjouralsorg 29 Page

5 The umber of verces ( p ( G )) he we have, vd deg( v ) 2 q Complemeary Tree Pared Domao Graphs V D s V D = p ad he umber of edges V D s p Sce G G 0G2 ad G have p p2 verces, here are a leas 2( p ) edges from V D o D Also deg ( ) v Hece, 2q ( p ) ( G ) (2( p ) Sce 2, we have 2 q p G) 2 2p 2 2 ( G ) 2q 2p 2 2 2p 2 2p q 4 III Complemeary Tree Pared Domao Number Ad Toal Domao Number of a Tree The followg upper bouds are kow o us Cockaye e al[2] oved ha, For ay coeced graph G of order 3, 2 / 3 Hayes e al[] oved ha, For ay graph G whou solaed verces, 2 Before o ove our ma resul, frs o ove he followg heorem Theorem 3: If v s a suppor verex of a coeced graph G, he v s every DOI: 09790/ wwwosrjouralsorg 30 Page se ad every se Proof: Le v be a suppor verex of a coeced graph G Suppose v s o a member of he complemeary ree pared domag se he he pede verex adjace o v s o domaed by ay oher verex of G So v mus be ese every se of G ad s also every se of G Observao 32:[5] For ay coeced graph G wh dameer a leas hree, here exss a -se ha coas o leaves of G For a verex u a rooed ree T, we deoe by T he sub ree of T duced by s descedes, s deoe suppor verces ad l deoe leaves We ow ese he upper bouds Theorem 33: If T s a o pah ree of order a leas fve wh equal umber of suppor verces ad leaves he ( T) ( T) ( T) s Proof: By he defo of CTPD- se he lower boud follows To ove he upper boud, we oceed hs heorem by duco o T whch s of order a leas fve wh s suppor verces ad l leaves I s obvous ha he equaly holds for p 5 Le dam (T) 5 Assume ha for ay ree T ' of order p, ' 6 p havg s suppor verces, ( T' ) ( T' ) s' Le T be a ree of order p wh s suppor verces, ad le ad D se ad a (T ) (T) se respecvely u S be a Roo he ree a a verex r of maxmum eccercy dam ( T) 5 Le u be a suppor verex a maxmum dsace from r ad v be a pare of u he rooed ree So deg( u ) = 2 Le w be he pare of v ad x be he pare of w By our choce of u, every chld of v s eher a leaf or a suppor verex of degree wo Now hree cases arses Case (): Le v has a chld besdes u say y ha s a suppor verex By heorem 3, u ad y are S Now eher () u may be pared wh s leaf u ' ad y may be pared wh s leaf y ' or () a mos oe of u or y s pared wh v I parcular le u be pared wh s leaf u ' ad y may be pared wh s leaf y ' Le

6 T ' Complemeary Tree Pared Domao Graphs T Sce y S, S T ' s a complemeary ree pared domag se of T', ad so T u ( T' ) ( T) 2 O he oher had, every (T ') -se ca be exeded o a complemeary ree pared domag se of T by addg u ad u ' Thus ( T ) ( T' ) 2 exss a (T ') se D' coag v ad y Hece 'u T ) ( T') every (T ) se coas v, y ad u, follows ha D V (T' ) s a TDS of ' ( T' ) ( T ) Thus ( T' ) ( T) ad s ' s By duco hypohess o ' By observaos 32, here D s a TDS of T ad ( Sce ( T' ) ( T' ) s' Therefore ( T) 2 ( T) ( s ) ad hece ( T) ( T) s We oceed he same ocedure for () Case (): Le v s a suppor verex ad has o chld besdes u of degree wo Le T' T T v T, ad so T, We have Assume ha T ' has order a leas hree The ( T) ( T' ) 2 ad s 2 s' s By heorem 3, u s D If N v) D w ) u ( u D, he u T' ) D s a TDS of T ' If N v) u D ( s a TDS of T ' Thus ( ( T) By duco hypohess o ', ( T' ) ( T' ) s' Therefore ( T) 2 ( T) ( s ) ad hece ( T) ( T) s Case () : Le v has o chld besdes u, ha s, deg( v ) 2Suppose deg( ) 3 ( he T We have w, ad T' T Tv The s ' s Ay (T' ) se ca be exeded o CTPD se of T by addg u ad ' u ad so, ( T) ( T' ) 2 Furhermore, by heorem 3, we may assume ha u ad v D If N ( w) v D, he D u, v ( D x) u, v s a TDS of ' s a TDS of T ' If N w) v D T Thus ( T' ) ( T) By duco hypohess o T', ( he We oba he desred equaly Ths heorem shows he relaoshp bewee oal domao umber ad complemeary ree pared domao umber of a ree Also shows ha he upper ad lower boud exss for complemeary ree pared domao umber of a ree erms of oal domao umber ad suppor verces of T IV Cocluso Ths paper sgfes he relaoshp bewee oal domao umber ad complemeary ree pared domao umber of a ree Also shows he upper ad lower boud exss for complemeary ree pared domao umber of a ree erms of oal domao umber ad suppor verces of T I fuure we have plaed o work for verex crcal ad edge crcal of he deermed parameer ad also we fd he relao bewee hem Refereces [] JA Body Murhy, Graph Theory wh Applcaos (Elsever scece publshg co, Ic, Newyork, NY007, 976) [2] EJ Cockaye, RM Dawes ad ST Hedeem(980), Toal domao graphs, Neworks, 0, 980, 2-29 [3] Jayapal Baskar Babujee, Babha Suresh, New Cosrucos of Edge Bmagc Graphs from Magc Graphs, Appled Mahemacs 2, 20, [4] VR Kull ad B Jaakram, The spl domao umber of a graph, Graph Theory Noes of New York, New York Academy of sceces XXXXII, (997), 6-9 [5] Musapha Chella ad Teresa W Hayes, Toal ad pared domao umbers of a ree, AKCE J Gaphs Comb, (2), (2004), [6] Musapha Chella ad Teresa W Hayes, O pared ad double domao graphs, Ulas Mah, 67, (2005, 6-7 [7] S Muhamma, M Bhaumah ad Vdya, Complemeary ree domao umber of a graph, Ieraoal Mahemacal forum, 6(26), (20), [8] O Ore, Theory of Graphs, Amer Mah Soc Colloq Publ, 38 Provdece, (962) [9] J Paulraj Joseph ad G Mahadeva, O complemeary perfec domao umber of a graph, Aca Ceca Idca, XXXIM(2), (2006), [0] T Tamzhchelvam ad Robso chellahura, Complemeary l domao umber of a graph, Tamkag Joural of Mahemacs, 40(2), (2009), [] Teresa W Hayes, Sephe T Hedem, Peer J Slaer, (Fudameals of Domao Graphs, Marcel Decker, 998) DOI: 09790/ wwwosrjouralsorg 3 Page