Intrnational Journal of Enginring Rsarch and Tchnology. ISSN 0974-3154 Volum 8, Numbr 1 (015), pp. 35-4 Intrnational Rsarch Publication Hous http://www.irphous.com Th Equitabl Dominating Graph P.N. Vinay Kumar Faculty of Mathmatics, Sri H.D.D. Govt. First Grad Collg, Hassan, Karnataka, INDIA Abstract Th quitabl dominating graph ED( G) of a graph G is a graph with V ( ED( G)) V ( G) D( G) whr DG ( ) is th st of all minimal quitabl dominating sts of G and u, v V ( ED( G)) ar adjacnt to ach othr if u V ( G ) and v is a minimal quitabl dominating st of G containing u. In this papr w charactriz th quitabl dominating graphs which ar ithr connctd or complt. Kywords: Minimum quitabl dominating st; Equitabl dominating graph; Minimum quitabl domination numbr. Mathmatics Subjct Classification (000): 05C 1. Introduction All th graphs ar simpl, undirctd without loops and multipl dgs. Lt G ( V, E ) b a graph. A subst D of V is said to b a quitabl dominating st of G if for vry v V D thr xists a vrtx u D such that uv E( G ) and d( u) d( v ) 1. Th minimum cardinality of such a dominating st D is calld th quitabl domination numbr of G and is dnotd by ( G ). An quitabl dominating st D is said to b minimal quitabl dominating st if no propr subst of D is an quitabl dominating st. Kulli and Janakiram [5] introducd a nw class of intrsction graphs. Motivatd by this w introduc a nw class of graphs in th fild of domination thory. Throughout this papr, th graph G is of p vrtics and q dgs. Th trms usd in this papr ar in th sns of Harary[4]. Dfinition 1.1[III]: A vrtx u V is said to b dgr quitabl with a vrtx v V
36 P.N. Vinay Kumar if d( u) d( v ) 1. A vrtx u V is said to b an quitabl isolat if d( u) d( v), v V. Dfinition 1.[III]: A minimal quitabl dominating st of maximum cardinality is calld st and its cardinality is dnotd by ( G ). Dfinition 1.3[III]: Lt u V. Th quitabl nighbourhood of u dnotd by N ( u ) is dfind as N ( u) { v V / v N( u), d( u) d( v ) 1}. Dfinition 1.4[III]: A subst S of V is calld an quitabl indpndnt st, if for any u S, v N ( u ) for all v S {} u. Th maximum cardinality of S is calld quitabl indpndnc numbr of G and is dnotd ( G ). Dfinition 1.5[III]: Th maximum ordr of a partition of V into quitabl dominating sts is calld quitabl domatic numbr of G and is dnotd by d ( G ). Dfinition 1.6: Th quitabl dominating graph ED( G) of a graph G is a graph with V ( ED( G)) V ( G) D( G) whr DG ( ) is th st of all minimal quitabl dominating sts of G and u, v V ( ED( G)) ar adjacnt to ach othr if u V ( G ) and v is a minimal quitabl dominating st of G containing u. An xampl of th quitabl dominating graph ED( G) of a graph G is givn blow:. Rsults In this sction w prov th main rsults of this papr. First w obtain th ncssary and sufficint condition for a givn graph G to b connctd and followd by som rsults on compltnss, quitabl domatic partition and th quitabl domination numbr of ED( G ). Thorm.1[III]: Lt G b a graph without quitabl isolatd vrtics. If D is a
Th Equitabl Dominating Graph 37 minimal quitabl dominating st, thn V D is an quitabl dominating st. Thorm.[I]: A graph G is Eulrian if and only vry of vrtx of G is of vn dgr. Thorm.3: For any graph G with p and without quitabl isolatd vrtics, th quitabl dominating graph ED( G ) of G is connctd if and only if ( G) p 1. Proof: Lt ( G) p 1. Lt D1 and D b two minimal quitabl dominating sts of G. W considr th following cass:- Cas i): Suppos thr xists two vrtics u D 1 and v D such that u and v ar not adjacnt to ach othr. Thn, thr xists a maximal quitabl indpndnt st D 3 containing u and v. Sinc vry maximal quitabl indpndnt st is a minimal quitabl dominating st, D3 is a minimal quitabl dominating st joining D1 and D. Hnc thr is a path in ED( G ) joining th vrtics of VG ( ) togthr with th minimal quitabl dominating sts of G. Thus, ED( G ) is connctd. Cas ii): Suppos for any two vrtics u D 1 and v D, thr xists a vrtx w D1 D such that w is adjacnt to nithr u not v. Thn, thr xists two maximal quitabl indpndnt sts D 3 and D4 containing u,w and w,v rspctivly. Thus, th vrtics u,v,w and th minimal quitabl dominating sts D1, D, D3, D 4 ar connctd by th path D1 u D3 w D4 v D. Thus, ED( G ) is connctd. Convrsly, suppos that ED( G ) is connctd. Lt us assum that ( G) p 1 and lt {} u b a vrtx of dgr p 1. Thn, { u} is a minimal quitabl dominating st of G and by thorm.1, V D has a minimal quitabl dominating st say D. This implis that ED( G) has at last two componnts, a contradiction. Hnc, ( G) p 1. Hnc th rsult. Rmark.4: In ED( G ), any two vrtics u and v of VG ( ) ar connctd by a path of lngth at most four. Thorm.5: For any graph G with ( G) p 1 and without quitabl isolatd vrtics, diam( ED( G )) 5. Proof: As ( G) p 1, by thorm.4, G is connctd. Lt ED( G) V Y, E, whr Y is th st of all minimal quitabl dominating sts of G. Lt u, v V Y. Thn, by abov thorm.4, diam( ED( G )) 4 if u, v V or u, v Y. On th othr hand, if u V and v Y thn v D is a minimal quitabl dominating st of G. If
38 P.N. Vinay Kumar u D, thn d( u, v ) 4; othrwis, thr xists a vrtx w D such that d( u, v) d( u, w) d( w, v ) 4 1 5. This provs th rsult. Thorm.6: For any graph G without quitabl isolatd vrtics, ED( G ) is a complt bipartit graph if and only if K p. Proof: Suppos that ED( G ) is not a complt bipartit graph with G K p. As G K p th minimal quitabl dominating st of G is VG, ( ) vry isolatd vrtx in ED( G ) is adjacnt to th vrtx VG. ( ) This implis that ED( G ) is K 1, p, which is a contradiction. Thus, ED( G) is complt bipartit graph. Convrsly, suppos that ED( G ) is complt bipartit graph and G K. Thus G contains a nontrivial subgraph G 1. Thn, by thorm.1, for som vrtx u G 1, thr xists a minimal quitabl dominating sts D and D with u D and u D, which is a contradiction to th fact that G is complt bipartit graph with u G 1. Hnc G K p. This complts th proof. Thorm.7: For any graph G without quitabl isolatd vrtics, d ( G) ( ED( G )). Furthr, th quality holds if and only if VG ( ) can b partitiond into union of disjoint minimal quitabl dominating sts of cardinality on. Proof: Lt S b th maximum ordr of quitabl domatic partition of VG. ( ) If vry quitabl dominating st is minimal and S consists of all minimal quitabl dominating sts of G, thn S is a maximum quitabl indpndnt sts of ED( G ). Hnc d ( G) ( ED( G )). Othrwis, lt D b a maximum quitabl indpndnt st with D S. Hnc, D is a minimal quitabl dominating st of G. Lt u D. Thn, thr ar two following cass:- Cas i): If u D', whr D' S. Thn, clarly S {} u is a quitabl indpndnt st in ED( G ). Hnc th rsult holds. Cas ii): If u D ', whr D' S. Thn, thr xists a vrtx w V ( G) such that S { u, w} is an quitabl indpndnt st. Hnc th rsult. Clarly, th quality condition follows as vry componnt of ED( G ) is K as VG ( ) is th union of disjoint minimal quitabl dominating sts of cardinality on. This complts th proof. Corollary.8: For any graph G, V ( ED( G) d ( G ). p
Th Equitabl Dominating Graph 39 Proof: Follows from thorm.8 and th fact that for any graph G, V ( G) ( G ). Thorm.9: For any graph G without quitabl isolatd vrtics p d ( G) p ' p( ( G ) 1), whr p ' is th numbr of vrtics of ED( G ). Furthr th lowr bound is attaind if and only if vry minimal quitabl dominating st of G is indpndnt and th uppr bound is attaind if and only if vry maximum quitabl indpndnt st is of cardinality on. Proof: Th graph ED( G) has th vrtx st V ( G) D( G) and it has at last d ( G) numbr of minimal quitabl dominating sts, hnc th lowr bound follows. Clarly uppr bound follows as vry maximal quitabl indpndnt st is a minimal quitabl dominating st and vry vrtx is prsnt in at most ( p 1) minimal quitabl dominating sts. Furthr, suppos that p d ( G) p '. As thr ar d ( G) numbr of minimal quitabl dominating sts and ach vrtx is prsnt in xactly on of th minimal quitabl dominating st and hnc ths minimal quitabl dominating sts ar indpndnt. Also, suppos that vry maximum quitabl indpndnt st is of cardinality on thn, ths ar minimal quitabl dominating sts of G and ar indpndnt and as vry vrtx is prsnt in at most ( p 1) minimal quitabl dominating st, th quality holds. This implis th ncssary condition. Convrs of th rsult trivially holds. Thorm.10: For any graph G without quitabl isolatd vrtics p d ( G) q ' p( p 1), whr q ' is th numbr of dgs of ED( G ). Furthr, th lowr bound is attaind if and only if vry minimal quitabl dominating st is indpndnt and th uppr bound is attaind if and only if G is ( p ) rgular. Proof: Th proof of th lowr bound follows by th sam lins of thorm.10. Suppos th lowr bound is attaind. As vry vrtx must b in xactly on of th dominating st, Clary vry minimal quitabl dominating st is indpndnt. As vry vrtx is in at most ( p 1) minimal quitabl dominating st, uppr bound follows. Suppos th uppr bound is attaind. Thn, ach vrtx is in xactly ( p 1) minimal quitabl dominating sts and hnc G is ( p ) rgular. This complts th proof. Thorm.11: For any graph G with p 3, d ( ED( G)) 1 if and only if G K p,
40 P.N. Vinay Kumar whr K p is th complmnt of K p or ED( G ) has an quitabl isolatd vrtx. Proof: Suppos that d ( ED( G )) 1. Thn, ED( G) has a vrtx D with D V ( G ). Thus ED( G ) is K 1, p and hnc G K p. Othrwis, suppos assum that ED( G) has p' no quitabl isolatd vrtx and V ( ED( G)) p '. Thn, ( ED( G )). If D is an quitabl dominating st, thn V D is an quitabl dominating st and hnc d ( ED( G )), a contradiction. Hnc ED( G) has an quitabl isolatd vrtx. Th convrs is obvious. Thorm.1: If a graph G is connctd, ( p 1) isolatd vrtics thn, ( ED( G)) p. rgular and without quitabl Proof: As G is connctd and ( G) p 1, by thorm.4, ED( G ) is disconnctd. Also, w know that vry vrtx is prsnt in at most ( p 1) minimal quitabl dominating sts. Thus, ED( G) is a disconnctd graph with ach of th componnt bing K, thr ar p numbr of componnts. Hnc ( ED( G)) p. Thorm.13: For any graph G of ordr p, without quitabl isolatd vrtics and ( G) p 1, th quitabl dominating graph ED( G) of a graph G is a tr if and only if G K p. Proof: As G is a graph of ordr p,without quitabl isolatd vrtics and ( G) p 1, by thorm.4, ED( G) is connctd. Suppos assum that ED( G) of G is a tr. Thn, clarly G has no cycl. On th contrary assum that G K p. Thn, by thorm.1, d ( ED( G )) 1. Hnc thr xists at last two minimal quitabl dominating sts containing whr u and v ar any two vrtics in G. If u and v ar in th sam minimal quitabl dominating st D thn, u D v u is a cycl in ED( G ), a contradiction. On th othr hand, if u and v ar in diffrnt minimal quitabl dominating st. Thn, thr xists vrtics u 1, v1 and th minimal quitabl dominating sts D 1, D and D3 such that uu1 D 1, u1v1 D and v1v D 3. Thus, u and v ar connctd by two paths in ED( G ), a contradiction. Hnc Convrsly, suppos that G K p and ( G) p 1. Thn, by thorm.4, ED( G) is connctd. Also, by thorm.1, d ( ED( G )) 1. i.., thr xists a minimal quitabl dominating st D with D V ( G ). Thus, ED( G) is connctd, K and has no cycl. Hnc ED( G ) is a tr. This complts th proof. 1, p Thorm.15: For any graph G, ED( G) is ithr connctd or has at most on G K p
Th Equitabl Dominating Graph 41 componnt that is not K. Proof: W considr th following cass:- Cas i): If ( G) p 1, thn by thorm.4, ED( G ) is connctd. Cas ii): If ( G) ( G) p 1, thn G K p. Hnc ach of th vrtx v V ( G) is a minimal quitabl dominating st of G and hnc ach of th componnt of ED( G ) is K. Cas iii): If ( G) ( G) p 1. Lt v1, v,..., vn b n vrtics of G of dgr p 1. Lt H G thn ( H) V ( H ) 1. Hnc by thorm.4, ED( H ) is { v, v,..., v } 1 n connctd. Sinc ED( G) ( V ( ED( H) V ( G1) V ( G)... V ( Gn)) whr G1, G,..., Gnar th graphs joining v1, v,..., vnwith { v1},{ v},...,{ v n } rspctivly. Thn, xactly on of th componnt of ED( G) is not K. Hnc th proof. Thorm.17: If G is a r rgular graph with ( G) and vry vrtx is in xactly vn numbr of minimal quitabl dominating sts thn ED( G) is ulrian. Proof: Lt G is a r rgular graph. Sinc ach of th vrtx of G is in vn numbr of minimal quitabl dominating sts, ach of thm contributs vn numbr to th dgr of th vrtx in ED( G ) and as ( G ), ach of th minimal quitabl dominating st of G is a vrtx of dgr two in ED( G ). Thus, by thorm., ED( G) is ulrian. Thorm.18: Lt G b a graph with ( G) p 1and ( G ). If vry vrtx is prsnt in xactly two minimal quitabl dominating sts thn, ED( G) is Hamiltonian. Proof: As ( G) p 1, G is connctd by thorm.4. Also, sinc vry vrtx is prsnt in xactly two minimal quitabl dominating sts, ( G) ( G) and also dg( u) dg( D ) in ED( G ), whr D is a minimal quitabl dominating st in G. Thus, ED( G) is connctd and -rgular. Hnc ED( G) is Hamiltonian. REFERENCES: I. Chartrand G. and Zhang P., Introduction to graph thory, Tata McGraw-Hill Inc., Nwyork (006). II. Cockayn E.J. and Hdtnimi S.T., Towards a thory of domination in
4 P.N. Vinay Kumar graphs, Ntworks, 7, 47-61 (1977). III. Dharmalingam K.M.,Studis in graph thory quitabl domination and bottlnck domination, Ph.D. Thsis (006). IV. Harary F., Graph Thory, Addison-Wsly Publ. Comp. Inc., Rading, Mass (1969). V. Kulli V.R. and Janakiram B., Th Minimal Dominating Graph, Graph thory nots of Nw York XXVIII, 1-15 (1995).