A Note on Chromatic Weak Dominating Sets in Graphs

Transcription

1 Iteratioal Joural of Mathematis Treds ad Tehology (IJMTT) - Volume 5 Number 6 Jauary 8 A Note o Chromati Weak Domiatig Sets i Graphs P. Selvalakshmia ad S. Balamurugab a Sriivasa Ramauja Researh Ceter i Mathematis, Sethupathy Govermet Arts College, Ramaathapuram, Tamiladu, Idia b PG Departmet of Mathematis, Govermet Arts College, Melur , Tamiladu, Idia a selvasarathi86@gmail.om, b balapoojaa9@gmail.om Abstrat I this paper hromati eak domiatig sets ad hromati eak domiatio umber are defied. Chromati eak domiatio umber for stadard graphs are foud, bouds for hromati eak domiatio umber are obtaied. AMS Subjet Classifiatio: 5C69 Keyords: Domiatio, Weak domiatio, Chromati eak domiatio, Chromati umber. Itrodutio Let G = (V, E) be a fiite, simple ad udireted graph ith either loops or multiple edges. The order ad size of G are deoted by ad m respetively. Oe of the fastest groig areas ithi graph theory is the study of domiatio ad related problems. A subset D of V is said to be a domiatig set of G if every vertex i V D is adjaet to a vertex i D. The miimum ardiality of a domiatig set is alled the domiatio umber of G ad is deoted by γ(g). A omprehesive treatmet of the fudametals of domiatio is give i the book by Hayes et al [7]. Prof. E. Sampathkumar ad L. Pushpa Latha itrodued the oept of strog (eak) domiatio i graphs i [9]. A set D V is alled a eak domiatig set if for every vertex v V D, there exists a vertex u D suh that uv E(G) ad deg(u) deg(v). The miimum ardiality of a eak domiatig set is alled the eak domiatio umber of G ad is deoted by γ (G). A eak domiatig set of ardiality γ (G) is alled as γ -set of G. For D V, This paper is a outome of the Major Researh Projet Sposored by DST (SERB) No: SB / EMEQ - 5 / ISSN: Page 464

2 Iteratioal Joural of Mathematis Treds ad Tehology (IJMTT) - Volume 5 Number 6 Jauary 8 the subgraph idued by D is deoted by < D >. A set D V is said to be a hromati preservig set if χ(< D >) = χ(g). It as defied by T. N. Jaakirama ad M. Poobalarajai [8]. A set D V is said to be a domhromati set if D is a domiatig set ad χ(< D >) = χ(g). The miimum ardiality of a dom-hromati set is alled the dom-hromati umber ad is deoted by γh (G). This oept is also defied by T. N. Jaakirama ad M. Poobalarajai [8]. A split graph is a graph G = (V, E) hose verties a be partitioed ito to sets V ad V, here the verties i V form a omplete graph ad the verties i V form a ull graph. A lique of a graph G is a maximal omplete subgraph. The ardiality of a maximum lique is alled the lique umber ad is deoted by ω(g). A graph G is a perfet graph if χ(h) = ω(h) for all idued subgraph H of G. The legth of a smallest yle (if ay) of a graph G is alled girth ad is deoted by g(g) ad the legth of a smallest odd yle (if ay) is deoted by g (G). A graph G is plaar if it a be embedded i a plae, a plae graph has already bee embedded i the plae. A set D V is said to be a hromati strog domiatig set if D is a strog domiatig set ad χ(< D >) = χ(g). The miimum ardiality of a hromati strog domiatig set is alled the hromati strog domiatio umber of G ad is deoted by γs (G). A hromati strog domiatig set of ardiality γs (G) is alled as γs -set of G. This oept as defied by V. Samiatha ad S. Balamuruga []. A graph G is alled a vertex χ-ritial graph if χ(g v) < χ(g) for all v V (G). A vertex v said to be a eak vertex if deg(v) < deg(u) for all u N (v). Defiitio.. [8] A subset D of V is said to be a hromati preservig set (or p-set) if χ(< D >) = χ(g). The miimum ardiality of a hromati preservig set i a graph G is alled the hromati preservig umber (or p-umber) of G ad is deoted by p(g). Chromati Weak Domiatig Sets Defiitio.. Let G = (V, E) be a graph. A subset D of V is said to be a hromati eak domiatig set (or d-set) if D is a eak domiatig set ad χ(< D >) = χ(g). The miimum ardiality of a hromati eak domiatig set i a graph G is alled the hromati eak domiatio (G). umber (or d-umber) ad is deoted by γ (G). Propositio.. Let G be a graph of order. The γ (G) = iff G = K. Propositio.. Let G be a graph. The γ ISSN: Page 465

3 Iteratioal Joural of Mathematis Treds ad Tehology (IJMTT) - Volume 5 Number 6 Jauary 8 Observatio.4. (i). Chromati eak domiatig set exists for all graphs. (ii). Vertex set V is a trivial hromati eak domiatig set. (iii). For a vertex-olor-ritial graph, V is the oly hromati eak domiatig set. (G). (iv). For ay graph G, p(g) γ (v). If D is a hromati eak domiatig set of G, the eah vertex of V D is ot adjaet to at least oe vertex of D. (vi). A d-set of a graph G is a global domiatig set. Proof. (i) to (iv) follos trivially. (v). Suppose D is a hromati eak domiatig set suh that x V D is adjaet to eah vertex of D. Sie D otais a p-set of G, let D D be a p-set of D. Sie x is adjaet to eah vertex of D, χ(g) χ(< D > ) + = χ(g) +, a otraditio. (vi). Let D be a d-set of a graph G. From (v), i G eah vertex of V D is adjaet to at least oe vertex of D. Hee D is a domiatig set of G. Sie D is a domiatig set of G, it follos that D is a global domiatig set. Results for Stadard Graphs Propositio.. Let G = P. The ( + if or ( mod ) γ (P ) = + if ( mod ) Proof. Let D be a γ -set of G. Case : Let ( mod ). Let = k. Sie γ (P ) = + if or ( mod ). Therefore k γ (P ) = + = k +. Therefore D = {u, u4, u7,..., uk, uk } is a uique γ -set of Pk. The χ(< D >) =, but χ(p ) =. Let D = D {u}, here u N (ui ), ui D. The χ(< D >) =. This implies D is a γ -set of Pk. Thus, γ (P ) = D = D {u} = D + = +. Therefore γ (P ) = + if ( mod ). Case : Let ( mod ). Let = k +. Sie γ (P ) = + if, ( mod ), γ (P ) = k+ + = k +. Let D = {u, u4, u7,..., uk, uk+, uk+ } be a uique γ -set of Pk+. Sie D is ot idepedet, χ(< D >) =. -set of P This implies D is a γ k+. Therefore γ (Pk+ ) = D = +. ISSN: Page 466

4 Iteratioal Joural of Mathematis Treds ad Tehology (IJMTT) - Volume 5 Number 6 Jauary 8 (P Therefore γ k+ ) = + if ( mod ). Case : Let ( mod ). Let = k +. Sie γ (P ) = if ( mod ), therefore γ (P ) = k+ = k +. Let D = {u, u4, u7,..., uk, uk+ } be a uique γ -set of Pk+. Sie D is ot idepedet, χ(< D >=, but χ(p ) =. Let D = D {u}, here u N (ui ), ui D. The χ(< D >) = -set of P. Thus γ (P ) = D = D + = +.. This implies D is a γ (P ) = + if ( mod ). Hee the proof. Therefore γ Propositio.. Let G = C. The if is odd γ (C ) = + if, ( mod ) ad is eve if ( mod ) ad is eve Proof. Let C be a yle of verties. Case : Let be odd. The χ(c ) = = χ(< D >). Therefore D is ot a proper subset of V (C ). -set of C. Therefore γ (C ) = V (C ) = D =. Hee Let D be a γ (C ) =. γ Case : Let be eve. Subase (a): Let ( mod ). Let = k +, k. Let D = {u, u5, u8,..., uk, uk } is a γ -set of C. Therefore set ad D is ot idepedet. Therefore D is a γ (C ) = γ (C ) =. γ Subase (b): Let ( mod ). Let = k, k. Let D = {u, u5, u8,..., uk }, D = {u, u6, u9,..., uk } ad D = {u, u4, u7,..., uk }. The D, D ad D are the oly γ -sets hih are also idepedets. Therefore, χ(< Di >) =, i =,, but (C ) > γ (C ). No, D {u } is a hromati eak χ(c ) = ad γ (C ) D + = γ (C ) + = +. domiatig set of C. Therefore, γ Subase (): Let ( mod ). k+ Let = k +, k. The γ (C ) = = = k +. Let D be a γ -set. Suppose D is ot idepedet. Let vj vj+ D. Cosider the path P : v j+, vj+4,..., v, v, v,..., vj of legth 4. Therefore, γ (P ) = 4 = k+ 4 = k = k. Therefore, γ (C ) = γ (P ) + = k +, hih is a otraditio. Therefore D is idepedet ad χ(g) =. The D {x} is a hromati eak domiatig set of C, x N (y) for some y D. Therefore, γ (C ) = γ (C ) + = +. (C ) = +. Hee γ 4 ISSN: Page 467

5 Iteratioal Joural of Mathematis Treds ad Tehology (IJMTT) - Volume 5 Number 6 Jauary 8 (K ) =. Propositio.. For, γ Proof. Let D be ay proper subset of V (K ). The χ(< D >) <. Therefore D is ot a hromati eak domiatig set. Therefore V (K ) is (K ) =. the oly hromati eak domiatig set. Hee γ (K ) =. Propositio.4. Let G = K. The γ Proof. Let G be totally disoeted graph. Therefore γ (K ) =. Thus = γ (K ) γ (K ). (K ) =. Therefore γ (K Propositio.5. Let G = K,,. The γ, ) =. Proof. Let D be a γ -set of K,. The D = ad χ(k, ) =, but χ(< D >) =. Sie eah vertex of D has degree ad let u be the vertex of K, ith degree. Let D = D {u}. This implies D = D + = + =. Therefore -set of K D = ad χ(< D >) =. Therefore D is a γ,. Hee γ (K, ) =,. Propositio.6. For m, ( γ (Km, ) = max{m, } + if m = if m = 6. Proof. Let G = Km,. Let D be a eak domiatig set of G. Case : Let m =. The γ (G) =. Let D = {u, v} here uv E(G). Sie G is bipartite, χ(g) =. Sie uv E(G), D is a hromati eak domiatig set of G (G) D =. Therefore γ (G) =. ad hee = χ(g) γ Case : Let m 6=. Let us osider to subases. Subase (a): Let > m. Let D = {u, u,..., u } be a miimal eak domiatig set of G. The χ(< D >) =, sie {u, u,..., u } is a idepedet set. But χ(g) =. Let D = D {vi }, here vi {u, u,..., um }. The χ(< D >) =. This -set of G. Therefore, γ (G) = D = D + = +. implies D is a γ (G) = γ (K Therefore γ m, ) = + if > m. Subase (b): Let < m. Proof of this ase is aalogous to the proof of subase (a). Hee the proof. 5 ISSN: Page 468

6 Iteratioal Joural of Mathematis Treds ad Tehology (IJMTT) - Volume 5 Number 6 Jauary 8 (D ) =, for all r s, Propositio.7. Let G = Dr,s. The γ r,s here = r + s +. Proof. Let D be a γ -set of Dr,s. The γ (Dr,s ) = r + s ad χ(dr,s ) =. But χ(< D >) =. Let deg(u) = r + ad deg(v) = s +. Sie r s, deg(u) deg(v). Let D = D {u}. The χ(< D >) = χ(< D {u} >) =. -set of D. Therefore, γ (D ) = D = D + = This implies D is a γ r,s r,s (D ) =. r + s + =. Therefore γ r,s Propositio.8. Let G = W. The ( if is eve γ (W ) = if is odd -set Proof. Let G = W. Let V (G) = {u, v, v,..., v }. Let D be a γ of G. Let u be a vertex ith deg(u) =. Case : Let be odd. The hv (G) {u}i is a eve yle. Therefore χ(hv (G) {u}i) =. Therefore χ(g) =. The D = {u, v, v,..., v } is a hromati eak (G) χ(< D >) =. Hee domiatig set. Therefore = χ(g) γ (G) = if is odd. γ Case : Let be eve. The < V (G) {u} > is a yle o verties. Sie is eve ad is odd. Therefore χ(< V (G) {u} >) = ad χ(g) = 4. Sie -set. Therefore χ(< V (G) {x} >) for all x V (G) sie D is a γ (G) = γ (W ) =. γ (F ) = + here F deotes a fa. Propositio.9. For, γ ( if, ( mod ). Proof. Sie γ (F ) = + if ( mod ) Let D be a eak domiatig set. Case : Let = k + ad = k +. Here χ(< D >) = beause o to verties of D are adjaet. Let D = D {u } here u u E(F ) ad u D. The χ(< D >) = χ(< D {u } >) =. But χ(f ) =. The u be the vertex of F ith degree. The D = D {u } otais -set of F. Therefore, a triagle ad χ(< D >) =. Thus D is a γ (F ) = D = D {u, u } = +. γ Case : Let = k. Here χ(< D >) = beause idued graph of D otais a edge. But χ(f ) =. Let D = D {u } here u is the vertex of F ith degree 6 ISSN: Page 469

7 Iteratioal Joural of Mathematis Treds ad Tehology (IJMTT) - Volume 5 Number 6 Jauary 8. The < D > otais a triagle ad χ(< D >) =. Therefore D -set of F. Thus γ (F ) = D = D {u } = + + = +. is a γ (F ) = +. Hee the proof. Therefore γ Theorem.. Give a positive iteger k, there exists a graph G suh (G) = k. that γ (K ) = k. Proof. Let G be a omplete graph Kk. The γ k 4 Bouds o γ -sets (G) = Propositio 4.. If G is a eve yle, the γ C8. iff G is C4, C6, Proof. Neessary oditio is trivial. (G) =. Coversely, suppose γ Case : Let ( mod ) ad be eve. (G) = +. The = 6. From Propositio., it is ko that γ Case : Let ( mod ) ad be eve. (G) =. The = 4. From Propositio., it is ko that γ Case : Let ( mod ) ad be eve. (G) = +4. The = 8. From Propositio., it is ko that γ (G) = γ (G) + iff ( mod Propositio 4.. If G is a path, the γ ). Proof. Neessary oditio is trivial. Coversely, suppose ( mod ) ad is eve. Let = k +. Let P : u, u, u,..., uk+ be a path o verties. Sie ( mod ), the γ (P ) =. D = {u, u4, u7,..., uk, uk+ } is a uique γ -set of P. This implies D is idepedet ad χ(< D >) = but χ(p ) =. Let D = D {ui }, here ui uj E(P ) ad uj D. The χ(< D >) =. -set of P. Thus γ (P ) = D = D {u } = D + = D is a γ i Therefore (P ) = γ (P ) +. Hee the proof. + = γ (P ) +. Hee γ Propositio 4.. Let D be ay d-set of G. The X V D deg(u). u D Proof. As D is a eak domiatig set, the result follos trivially. 7 ISSN: Page 47

8 Iteratioal Joural of Mathematis Treds ad Tehology (IJMTT) - Volume 5 Number 6 Jauary 8 Propositio 4.4. Let D be ay d-set of G. The V D = iff G = pk, p. P u D deg(u) Proof. If G = pk, the D = V ad deg(u) P = for eah u D. The the equality holds. No, suppose V D = u D deg(u) = k. Claim: k =. Suppose k. The to ases arise. Case : G is oeted. The χ(g). Let V D = {u, u,..., uk }. Sie D is a domiatig set eah ui is adjaet to a vertex of D ad hee, otributes at least oe degree to D. Sie χ(< D >) P, D otais at least oe edge hih otributes degrees to D. Hee, u D deg(u) k +, hih is a otraditio. Case : G is disoeted. If G is totally disoeted, V = D ad hee, V D = k =, a otraditio. Hee G has o trivial ompoet ad the < D > otais at least oe edge. The by a similar argumet as i ase (), otraditio arises. I both the ases, P otraditio arises. Therefore k =. The V D = u D deg(u) =. Therefore V = D ad hee, for eah u V, deg(u) =. Thus, G is a totally disoeted graph ad hee G = pk. Corollary 4.5. For ay o trivial oeted graph ith a d-set D, X deg(u) V D +. u D P Proof. If G is vertex olor ritial, the V = D ad u D V V D +. Suppose G is ot vertex olor ritial. As G is o-trivial, χ(g). By similar argumet as i ase () of the above propositio, P u D deg(u) V D +. Theorem 4.6. If G is a plaar graph ith diam(g) =, χ(g) = ad (G) 5. γ (G) =, the γ Proof. Loer boud is trivial. Let S = {a, b} be a γ -set of G. Sie diam(g) =, g (G) = or 5. Case : g (G) =. Let C be a -yle xyzx. If a, b / C, the verties of C are adjaet to a ad oe vertex is adjaet to b or vie versa, otherise K4 is idued, a otraditio. Let x ad y be adjaet to b. The axya is a -yle. Hee {a, x, y, a} is a d-set of G. If a or b is i the -yle together ith the 8 ISSN: Page 47

9 Iteratioal Joural of Mathematis Treds ad Tehology (IJMTT) - Volume 5 Number 6 Jauary 8 remaiig vertex of S is a d-set of G. Case : g (G) = 5. Let C be a 5-yle uvxyu. If a, b / C, the as S is domiatig, verties of C are adjaet to a ad oe vertex is adjaet to a or b ad ot to both, otherise -yle is idued. Also o to oseutive verties of C a be both adjaet to a or b. Otherise a -yle is idued. The S a domiate at most 4 verties of C, a otraditio. Hee a or b C. Let a C ad b / C. Let u = a. The x ad are adjaet to b ad hee a -yle is idued, a otraditio. Therefore -set of G. From ase () ad (), the both a, b C ad hee C is a γ upper boud is proved. Theorem 4.7. A hromati eak domiatig set D is miimal iff for eah u D oe of the folloig oditio hold: (i). χ(< D {u} >) < χ(g) (ii). u is a eak isolate of D (iii). there exists s V D suh that N (s) D = {u}. Proof. Let a hromati eak domiatig set D be miimal. The D {u} is ot a hromati eak domiatig set. This implies either D {u} is ot a hromati eak domiatig set or χ(< D {u} >) 6= χ(g). If χ(< D {u} >) 6= χ(g), learly χ(< D {u} >) < χ(g). Suppose D {u} is ot a hromati eak domiatig set, there exists s V (D {u}) suh that s is ot eak domiated by ay vertex of D {u}. If = u, N (u) D = φ, that is, u is a eak isolate of D. Let / u. The V D. This implies is strog domiated by u. N () D = {u}. Hee the proof. (G). Theorem 4.8. If G is a vertex-olor-ritial, the α (G) < γ (G) =. Let u V (G). Let S = V {u}. The S is a Proof. Suppose γ (G). vertex over of G. Therefore, α (G) S = V {u} = < = γ Therefore α (G) < γ (G). Propositio 4.9. If G is vertex-olor-ritial graph ith diam(g), the (G). α (G) + γ Proof. Give diam(g). The there exists u, v V (G) suh that uv / E(G). Let S = V {u, v}. The S is a vertex over of G. Therefore, (G). Therefore α (G) + α (G) S = V {u, v} = = γ γ (G). (G) 5. Propositio 4.. If G is a triagle free ith χ(g), the γ 9 ISSN: Page 47

10 Iteratioal Joural of Mathematis Treds ad Tehology (IJMTT) - Volume 5 Number 6 Jauary 8 Proof. Sie χ(g), ay d-set of G otais a odd yle. Sie G is (G) 5. triagle free, γ Propositio 4.. If H is a spaig subgraph of G suh that χ(h) = (G) γ (H). χ(g), the γ 5 Graphs ith respet to (G)+ (G) = + Theorem 5.. Let G be a graph ith (G) =. The γ iff G = K ( )K. (G) =. Therefore Proof. If G = K ( )K, the (G) = ad γ γ (G) + (G) = +. Coversely, suppose G is a graph ith (G) = satisfyig that (G) + (G) = +. The eah ompoets of G is either K or K γ (G) =. If there are more tha ith at least oe K ompoet ad γ oe K ompoet, the G a be hromati eak domiated by less tha verties. Hee G has exatly oe K ompoet ad every other ompoet is K. Therefore G = K ( )K. (G) + (G) = Theorem 5.. Let G be a graph ith (G) =. The γ iff G = K ( 4)K. (G) =. Proof. If G = K ( 4)K, the (G) = ad γ Therefore γ (G) + (G) = + =. Coversely, suppose G is a graph ith (G) = satisfyig that γ (G) + (G) =. The eah ompoets of G is either K or K ad (G) =. If there are more tha to K ompoets the G a be γ hromati eak domiated by verties. Hee G has exatly to K ompoets ad every other ompoet is K. Therefore G = K ( 4)K. Propositio 5.. For ay positive iteger k, there exists a vertex olor (G) α (G) = k. ritial graph G suh that γ (G) = k + Proof. Let = k +. Cosider a graph G = C. The γ k+ (G) α (G) = k + (k + ) = k. ad α (G) = = k +. Thus γ Hee the proof. Theorem 5.4. Give a positive iteger k, there exist a graph G suh that (G) γ (G) = k. (i). γ (G) γ (G) = k. (ii). γ s ISSN: Page 47

11 Iteratioal Joural of Mathematis Treds ad Tehology (IJMTT) - Volume 5 Number 6 Jauary 8 (G) = Proof. (i) Let G be a omplete graph ith k + verties. The γ k + ad γ (G) =. Therefore, γ (G) γ (G) = k + = k. (G) = k + ad γ (G) =. Therefore, (ii) Let G = Kk+. The γ s (G) γ (G) = k + = k. γ s Theorem 5.5. Give a positive iteger k, there exists a graph G suh that (G) γ (G) = k. (i). γ s (G) γ (G) = k. (ii). γ h (G) = k + ad γ (G) =. Therefore Proof. (i). Let G = K,k+. The γ s (G) γ (G) = k + = k. γ s (G) γ (G) = (ii). Let G = K,k+. The γh (G) =. Therefore γ h k + = k. 6 Fratioal Chromati Weak Futio Defiitio 6.. Let G = (V, E) be a graph. Let g : V (G) [, ] be suh that g(n [v] ) for all v V (G) ad the fratioal hromati umber of ({v V : g(v) > }) is the same as the fratioal hromati umber of G. The g is alled a fratioal hromati eak futio. The miimum eight of suh a futio is alled fratioal hromati eak umber of G ad is (G). deoted by γf Observatio 6.. (i). γ (K ) =. (K ) =. (ii). γ (iii). γf (K ) =. (K ) =. (iv). γf Referees [] S. Balamuruga, P. Aristotle ad V. Samiatha, Chromati Strog domiatig sets i Bipartite graphs, AKCE Iteratioal Joural of Graphs ad Combiatoris, Submitted. [] S. Balamuruga, G. Prabakara ad V. Samiatha, O Chromati Strog Domiatig Sets i Graphs, Iteratioal Joural of Egieerig Siee, Advaed Computig ad Bio - Tehology, Vol. Issue (), ISSN: Page 474

12 Iteratioal Joural of Mathematis Treds ad Tehology (IJMTT) - Volume 5 Number 6 Jauary 8 [] S. Balamuruga, A. Wilso Baskar ad V. Samiatha, Equality of strog domiatio ad hromati strog domiatio i graphs, Iteratioal Joural of Mathematis ad Soft Computig, Vol.. No. (), [4] Dieter Gerert, Iequalities betee the domiatio umber ad the hromati umber of a graph, Disrete Math. 76 (989) 5-5. [5] F. Harary, Graph Theory, Addiso Wesley, readig Mass (97). [6] J. H. Hattigh ad M. A. Heig, O strog domiatio i graphs, J. Combi.Math. Comb. Comput. 6 (998) 7-8. [7] T. W. Hayes, S. T. Hedetiemi ad P. J. Slater, Fudametals of Domiatio i Graphs, Marel Deker I.. Ne york, 998. [8] M. Poobalarajai, O Some Colorig ad Domiatio Parameters i Graphs, Ph.D Thesis, Bharathidasa Uiversity, Idia, 6. [9] E. Sampathkumar, L. Pushpa Latha, Strog eak domiatio ad domiatio balae i a graph, Disrete Mathematis, 6 (996), 5-4. [] P. Thagaraju, Studies i Domiatio i graphs ad Degree sequees, Thesis, Nov., (). ISSN: Page 475