The inverse strong non-split r-domination number of a graph

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1 MultiCaft Intenational Jounal of Engineeing, Science and Technology INTERNATIONAL JOURNAL OF ENGINEERING, SCIENCE AND TECHNOLOGY MultiCaft Limited. All ights eseved The invese stong non-split -domination numbe of a gaph B.K. Ameenal 1 *, R. Selvakuma 2 1 Depatment of Mathematics, Dhanabagiyam Kishnaswamy Mudalia College fo Women (Autonomous), Velloe 1, INDIA 2 Depatment of Mathematics, VIT Univesity, Velloe 14, INDIA (ameena_maths@ediffmail.com (B.K. Ameenal); * Coesponding autho) Abstact In this pape, we define the notions of invese stong non-split -dominating set and invese stong non-split -domination numbe γ sns (G) of a gaph G. We chaacteize gaphs fo which γ sns (G) + γ sns (G) = n, whee γ sns (G) is the stong non-split -domination numbe of G. We get many bounds on γ sns (G). Nodhaus-Gaddum type esults ae also obtained fo this new paamete. Keywods: -independent set, -dominating set, stong non-split -dominating set, invese stong non-split -dominating set, stong non-split -domination numbe and invese stong non-split -domination numbe. 1. Intoduction Gaphs seve as mathematical models to analyse successfully many concete eal-wold poblems. In pesent time, gaphs ae used to solve poblems in many fields. Thee is vitually no end to the list of poblems that can be solved with gaph theoy. We can use this theoy to a geat advantage in infomation etieval, physics, chemisty, genetics, psychology, achitectue, linguistics, sociological stuctues, compute technology, economics and so on. The simple fact that gaph theoy seves as a mathematical model fo any system involving a binay elation needs to be noted. Gaphs have an intuitive and aesthetic appeal because of thei diagammatic epesentation. Gaphs can be used to detemine whethe a cicuit can be implemented on a plana cicuit boad. We can distinguish between two chemical compounds with the same molecula fomula but diffeent stuctues, using gaphs. Gaphs can be used to study the stuctue of Wold Wide Web (Ameenal and Selvakuma, 2009). We can detemine whethe two computes ae connected by a communication link using gaph models of compute netwok. Gaphs with weights assigned to thei edges can be used to solve poblems such as the shotest path between two cities in a tanspotation netwok. We can also use gaphs to schedule examinations and assign channels to television stations. Gaphs ae used as models to epesent the competition of diffeent species in an ecological niche, in computing the numbe of diffeent combinations of flights between two cities in an ailine netwok, in finding the numbe of colous need to colou the egions of a map. Gaph theoy can be applied effectively to study switching netwoks. Switching theoy came into being with the publication of Eheniest s pape in 1910, in which he suggested that Boolean algeba could be applied to the oganization of telephone exchanges. In ecent yeas, the enomous gowth of switching theoy has been mainly motivated by its use in the design of digital computes. One of the easons fo the ecent evival of the inteest in gaph theoy among the students of electical engineeing is the application of gaph theoy to the analysis and design of electical netwoks, moe commonly known as electical cicuits. The idea of using gaph theoy in electical netwok is not new, as G. Kichhoff (1847) and JC Maxwell (1892) used them. Geate stides ae made in ecent times due to the aival of high speed digital compute. A milestone in gaph theoetic analysis of electical netwoks was achieved by W.S. Pecival (Deo, 2003). Compute pogammes ae now available fo analysis of lage netwoks based on gaph theoetic appoach. It is to be noted that in electical engineeing, the tem banch is used fo edge, node fo vetex and loop fo cicuit. Gaph theoy is a vey natual and poweful tool in Combinatoial OR (COR). We have aleady touched upon taveling salesman poblem and locating the shotest path between the two vetices in a gaph. We came acoss thee impotant classes of poblems in COR; the tanspotation poblems, activity netwok and game theoy. These poblems can be fomulated and solved elegantly as

2 128 Ammeenal and Selvakuma / Intenational Jounal of Engineeing, Science and Technology, gaph theoy poblems involving, connected and weighted digaphs. Fom a pactical point of view, all these poblems ae tivial if the netwok is small. Many eal life situations howeve consist of huge netwoks and theefoe it is impotant to look at these netwok poblems in tems of solving them using computes. Thee ae moe than 75 models of dominating and elative types of sets in gaphs, which have appeaed in the eseach liteatue fo the past 30 yeas (Haynes et al., 1998a,b). Many domination paametes ae fomed by combining domination with anothe gaph theoetic popety P. In an attempt to fomalize the concept, Haynes et al. defined the conditional domination numbe γ(g:p) as the smallest cadinality of a dominating set D V such that the induced subgaph <D> satisfies the popety P. Some of them ae: P 1 : <D> has no edges P 2 : <D> has no isolated vetices P 3 : <D> is connected P 4 : <D> is independent P 5 : <D> has a Hamiltonian cycle and so on. Clealy γ(g) γ(g:p) fo any popety. Note that with the exemption of independent domination, these conditional domination paametes do not exist fo all gaphs. Any undefined tem in this pape may be found in Haynes et al. (1998). Cockayne and Hedetniemi (1977) intoduced total domination in gaphs. Sampathkuma and Walika (1979) intoduced the concept of connected domination in gaphs. A set D of vetices in a gaph G = (V,E) is called a dominating set if evey vetex in V-D is adjacent to some vetex in D. The dominating set D V is a total dominating set if the induced subgaph <D> has no isolated vetices. The dominating set D V is a connected dominating set if the induced subgaph <D> is connected. The connected domination numbe γ c (G) is the minimum cadinality of a connected dominating set. Kulli and Janakiam (1997, 2000) intoduced the concept of split and non-split domination in gaphs. A dominating set D V of a gaph G is a split (non-split) dominating set if the induced subgaph <V-D> is disconnected (connected). The split (non-split) domination numbe γ s (G) (γ ns (G)) is the minimum cadinality of a split (non-split) dominating set. Ameenal and Selvakuma (2008) intoduced the concept of Invese split and non-split domination numbes in gaphs. Let D be the minimum Invese dominating set of G w..to. D. Then D is called an Invese split (non-split) dominating set of G if the induced subgaph <V-D > is disconnected (connected). The invese split (non-split) domination numbe is denoted by γ s (G) (γ ns (G)) and it is the minimum cadinality taken ove all the minimal invese split (non-split) dominating sets of G. Let G = (V, E) be a finite, connected and undiected gaph without loops o multiple edges. Let be a positive intege. A non-empty subset D V is a -dominating set if evey vetex in V-D is within a distance fom atleast one vetex of D. The -domination numbe γ (G) is the minimum cadinality taken ove all the minimal -dominating sets of G. A -dominating set D V is said to be non-split -dominating set if the induced subgaph <V-D> is connected. The non-split -domination numbe γ ns (G) is the smallest cadinality of a non-split -dominating set of G. A non-split -dominating set is said to be a stong non-split -dominating set if the induced subgaph <V-D> is -complete. The stong non-split -dominating numbe γ sns (G) is the minimum cadinality of the stong non-split -dominating set of G. Let D be the minimum -dominating set of G. If V-D contains a -dominating set D then D is called an Invese -dominating set of G. Kulli and Sigakanti (1991) intoduced invese domination in gaphs. Wheneve D is a dominating set, V-D is also a dominating set. Let D be the minimum dominating set of G. If V-D contains a dominating set D then D is called an Invese dominating set of G with espect to D. In this pape, we define the Invese stong non-split -dominating set and the Invese stong non-split - domination numbe of a gaph. The -dominating set D is called the Invese stong non-split -dominating set if the induced subgaph <V-D > is connected and -complete. The Invese stong non-split -domination numbe (denoted by γ sns (G)) is the cadinality of the smallest invese stong non-split -dominating set of G. In this pape, we chaacteize gaphs fo which γ sns (G) + γ sns (G) = n. We get many bounds on γ sns (G). Nodhaus-Gaddum type esults ae also obtained fo this new paamete. Kulli and Janakiam (2003) also intoduced the concept of stong non-split domination numbe in gaphs. A dominating set D of a Gaph is a stong non-split dominating set, if the induced subgaph <V-D> is complete and connected. The stong non-split domination numge is denoted by γ sns (G) and it is the minimum cadinality of a stong non-split dominating set. Timothy et al. (1994) intoduced the concept of distance domination. When distance domination is intoduced, the elements of the complement of a dominating set need not have diect links with elements of the dominating set. Even if the elements ae at some distance fom the elements of a dominating set, the communication may be still possible. Fo example, when using adio fequencies, one can sent the communication to the ecipients within the ange of cetain adius. The idea of -domination demands that thee should be atleast one peson in the dominating set at a distance less than o equal to fom any given peson in the complement. To manage situations of impossible Fault toleance in the dominating sets, we impose conditions that the complement of the dominating set is intact. That is, the communication fom one element of the complement to anothe may not take moe time. In othe wods, the complement is connected and the maximum distance between any two elements is also less than o equal to. Fo Gaph peliminaies, we efeed to Oe (1962) and Chatand and Lesniak (1996). Fo the popeties of complementay gaphs, we efeed to Chatand and Schuste (1974).

3 129 Ammeenal and Selvakuma / Intenational Jounal of Engineeing, Science and Technology, 1.1 Note The distance between any two vetices u, v V(G) is the length of the shotest path joining them and it is denoted by d(u,v). The diamete of a gaph G is the maximum distance between any two vetices in V(G) and it is denoted by diam(g). A non-empty subset D V is said to be independent if no two vetices in D ae adjacent. The open -neighbouhood N (v) of a vetex v in a gaph is defined by N (v) = {u V / 0 < d(u,v) } and its closed -neighbouhood is N (v) = N (v) {v}. The -degee of a vetex v in G is denoted by deg (v) and is given by N (v). Let Δ (G) and δ (G) denote the maximum and minimum -degee among all the vetices of G espectively. Let v D. Define N D (v) = N (v) D. Let Δ (G) = max N D (v) and δ (G) = min N D (v) D is a -independent set if Δ <D>=0 The -independence numbe β (G) is the maximum cadinality of an -independent set. 2. Results and Bounds Hee we obseve the exact values of γ sns (G) fo some standad gaphs which ae staightfowad and pove some standad esults. 2.1 Obsevation and Results Obsevation (i) Fo any complete gaph K n with n 2 vetices γ sns (K n ) = 1 fo all 1 (ii) Fo any complete bipatite gaph K m,n 2 m n γ sns (K m,n ) = 0 if Obsevation Fo any gaph G, γ (G) γ ns (G) γ sns (G) Definition (Pushpalatha, 2009) A subset D V is a -dominating set if evey vetex in V-D is within a distance fom at least one vetex of D. The minimum cadinality of a -dominating set is called the -domination numbe of G and is denoted by γ (G). D V is a non-split -dominating set of G if D is a -dominating set and the induced subgaph <V-D> is connected. The minimum cadinality of a non-split -dominating set of G is called a non-split -domination numbe of G and is denoted by γ ns (G). Two vetices u, v V(G) ae said to be -adjacent if d(u,v). A gaph G is said to be -complete if evey vetex in V(G) is -adjacent to evey othe vetex in V(G). A -dominating set D V is said to be a stong non-split -dominating set if the induced subgaph <V-D> is -complete and connected. The minimum cadinality of a stong non-split -dominating set is called a stong non-split -domination numbe and is denoted by γ ns (G) Definition Let D be the minimum -dominating set of G. If V-D contains a -dominating set D then D is called an Invese -dominating set w..to D. The Invese -domination numbe of G is denoted by γ (G) and it is the cadinality of the smallest invese -dominating set of G. The Invese -dominating set D is called the invese non-split -dominating set if the induced subgaph <V- D > is connected. D is called the Invese stong non-split -dominating set if the induced subgaph <V-D > is -complete and connected. The invese stong non-split -domination numbe is denoted by γ sns (G) and it is the cadinality of the smallest invese stong non-split -dominating set of G Note A subset D V-D is a -dominating set if evey vetex in V-D is within a distance fom at least one vetex of D Theoem Fo any spanning subgaph H of G, γ sns (G) γ sns (H) Poof: Let D be an invese stong non-split -dominating set of G. Then the induced subgaph <V(G)-D > is connected and -complete. Since V(H) = V(G), the induced subgaph <V(H)-D > is also connected and -complete. Theefoe, D is also an invese stong

4 130 Ammeenal and Selvakuma / Intenational Jounal of Engineeing, Science and Technology, non-split -dominating set of H. Hence evey invese stong non-split -dominating set of G is an invese stong non-split - dominating set of H. Hence γ sns (G) γ sns (H) Theoem An invese stong non-split -dominating set D of G is minimal if and only if fo all v D, one of the following conditions holds: (i) thee exists a vetex u V-D such that N (u) D = {v} (ii) thee exists a vetex w V-D such that w is not a distance with v. Poof: Let D be an invese stong non-split -dominating set of G. Assume that D is minimal. Theefoe, D -{v} is not an invese stong non-split dominating set fo any v D. Now to pove that anyone of the above two conditions is satisfied. On the contay, if thee exists a vetex u D such that u does not satisfy any of the given conditions then D = D -{u} is an invese dominating set of G. Also the induced subgaph <V-D > is connected and -complete. This implies that D is an invese stong non-split dominating set of G which contadicts the minimality of D. This poves the necessay pat. Convesely, fo any connected gaph G, if anyone of the two given conditions is satisfied, we get the condition is sufficient Definition Fo any intege 1, a gaph G = (V, E) is said to be -complete if evey vetex in V(G) is -adjacent to evey othe vetex in V(G). Then -clique is denoted by ω (G) and is defined by the maximum ode of -complete vetices. 2.2 Bounds on γ sns (G) Hee, we get bounds on γ sns (G) though the following theoems Theoem Fo any gaph G, n-ω (G) γ sns (G) n-ω (G)+1 Poof: Let D be a γ sns (G) set of G. Since the induced subgaph <V-D > is -complete, ω (G) V-D Let S be a -complete gaph with S = ω (G). Then fo any vetex u S, (V-S) {u} is an invese stong non-split -dominating set of G. Hence γ sns (G) n-ω (G) Theoem Let G be a gaph with ω (G) δ (G) Then γ sns (G) n-δ (G) Poof: Suppose ω (G) δ (G) +1 then using theoem 2.2.1, γ sns (G) n-δ (G) Let ω (G) = δ (G). Let D be a vetex in V-D. Then V-D is a -dominating set. Hence γ sns (G) V-D = n-ω (G) = n-δ (G) Thus γ sns (G) n-δ (G) Note: Fo any tee T, δ (T) = 1 Hence γ sns (G) n-1 ω set of G with D = ω (G). Then evey vetex in D is at a distance with at least one

5 131 Ammeenal and Selvakuma / Intenational Jounal of Engineeing, Science and Technology, 3 Chaacteization In this section, we chaacteize gaphs fo which γ sns (G)+ γ sns (G) = n Theoem Let G be a connected gaph with n 3 and δ (G)= 1 Let L V be the set of all leaves (one degee vetices) and let S=N(L) (stems). Then γ sns (G)+ γ sns (G) = n if and only if the following two conditions hold: (i) V-S is an -independent set (ii) Fo evey vetex x V-(S L), evey stem in N(x) is -adjacent to at least two leaves Poof: Suppose γ sns (G) + γ sns (G) = n Case (i) Let V (S L) = φ In this case, V S = L is a -independent set and both conditions hold Case (ii) Let V (S L) φ We will fist show that V-S is a -independent set. If not, let N 1 = N(S) (S L) and N 2 = V (N 1 S L) Claim: N 2 φ Assume N 2 is empty. Thee is an edge uv in <N 1 >. Clealy S is a γ sns set and N 1 L {u} is a γ sns set. Hence γ sns (G) + γ sns (G) n-1, a contadiction. Thus N 2 φ. Let D be a γ sns - set with D = S and D be a γ sns set γ sns (G) L + V (S L) = V-S (1) Since evey invese stong non-split -dominating set lies outside a stong non-split -dominating set, γ sns (G) V-S (2) Fom (1) and (2) we get γ sns (G) = V-S Theefoe, we have γ sns (G) + γ sns (G) = n 4. Nodhaus Gaddum type esults (Nodhaus and Gaddum, 1956) Theoem Let G and G be connected complementay gaphs with δ (G) = 1. Then (i) γ sns (G) + γ sns ( G ) 2(n-1) (ii) γ sns (G). γ sns ( G ) (n-1) 2 Poof: By theoem 2.2.2, γ sns (G) n - δ (G) Hee δ (G) = 1 γ sns (G) n-1 Similaly γ sns ( G ) n - δ ( G ) δ (G) = 1 δ ( G ) = 1 Hence γ sns ( G ) n 1 Hence the esult. 5. Conclusions Gaph theoy seves as a model fo any binay elation. In domination, both dominating sets and thei inveses have impotant oles to play. Wheneve, D is a dominating set, V-D is also a dominating set. In an infomation etieval system, we always have a set of pimay nodes to pass on the infomation. In case, the system fails, we have anothe set of seconday nodes, to do the job in the complement. When the complement set is connected, then thee will be flow of infomation among the membes of the

6 132 Ammeenal and Selvakuma / Intenational Jounal of Engineeing, Science and Technology, complement. Thus, the dominating sets and the elements in the invese dominating sets can stand togethe to facilitate the communication pocess. They play vey vital ole in coding theoy, compute science, opeations eseach, switching cicuits, electical netwoks etc. Thus in this pape, we defined the Invese stong non-split -dominating set and the Invese stong nonsplit -domination numbe of a gaph. We also chaacteized gaphs fo which γ sns (G) + γ sns (G) = n. We obtained many bounds on γ sns (G). Nodhaus-Gaddum type esults wee also obtained fo this new paamete. Nomenclatue G Simple gaph V Vetex set E Edge set <D> Subgaph induced by a set D N(v) Open neighbouhood of v N[v] Closed neighbouhood of v γ(g) Domination numbe γ (G) Invese domination numbe K m,n Bipatite gaph with m+n vetices γ ns (G) Non-split domination numbe γ ns (G) Invese non-split domination numbe K n Complete gaph with n vetices γ ns (G) non-split -domination numbe γ ns (G) Invese non-split -domination numbe γ sns (G) Stong non-split -domination numbe γ sns (G) Invese stong non-split -domination numbe γ (G) -domination numbe ω (G) -clique, maximum ode of -complete vetices δ(g) minimum degee among the vetices in G Δ(G) maximum degee among the vetices in G δ (G) minimum -degee among all the vetices in G Δ (G) maximum -degee among all the vetices in G Refeences Ameenal B.K. and Selvakuma, R The Invese split and non-split domination numbes in gaphs. Poc. of the Intenational Confeence on Mathematics and Compute Science, ICMCS 2008, Dept. of Mathematics, Loyola College, Chennai July 25-26, pp Ameenal B.K. and Selvakuma, R The invese stong non-split -domination numbe of a gaph. Poc. of the National Confeence on Industial Applications of Mathematics, NCMA 2009, PG and Reseach Dept. of Mathematics, Saced Heat College (Autonomous) Tiupattu, Velloe Dist., Mach 12-13, pp Chatand G. and Lesniak L Gaphs and Digaphs. Chapman and Hall / CRC, Boca Raton, FL. Chatand G. and Schuste S On the independence numbe of complementay gaphs. Tans. of the New Yok Acad. of Sci. Seies II, Vol. 36, No. 3, pp Cockayne, E.J. and Hedetniemi S.T Towads a theoy of domination in gaphs. Netwoks, Vol. 7. pp Deo N Gaph Theoy with Application to Engineeing and Compute Science. Pentice Hall of India, New Delhi. Haynes, T.W., Hedetniemi S.T. and Slate P.J. 1998a. Domination in Gaphs: Advanced Topics, Macel Dekke Inc. New Yok, U.S.A. Haynes, T.W., Hedetniemi S.T. and Slate P.J. 1998b. Fundamentals of domination in gaphs, Macel Dekke Inc. New Yok, U.S.A. Pushpalatha A.P., Suganthi S., Jothilakshmi G., Swaminathan V., Stong non split -domination numbe of a gaph. Electonic Notes in Discete Mathematics, Vol. 33, pp Kulli, V.R. and Janakiam B The split domination numbe of a gaph. Gaph Theoy notes of New Yok. New Yok Academy of Sciences, XXXII. pp Kulli, V.R. and Janakiam B The non-split domination numbe of a gaph. The Jounal of Pue and Applied Math. Vol. 31, No. 5, pp Kulli, V.R. and Janakiam B The stong non-split domination numbe of a gaph. Intenational Jounal of Management and Systems. Vol. 19, No. 2, pp

7 133 Ammeenal and Selvakuma / Intenational Jounal of Engineeing, Science and Technology, Kulli, V.R and Sigakanti.S.C Invese domination in gaphs. National Academy Science Lettes, Vol. 14, No. 12, pp Nodhaus, E.A. and Gaddum J.W On complementay gaphs. Ame.Math.Monthly, Vol. 63. pp Oe, O Theoy of Gaphs. Ameican Mathematical Society Colloq. Publ., Povidence, RI, 38. Sampathkuma, E. and Walika H.B The connected domination numbe of a gaph. J.Math.Phy.Sci., Vol. 13. pp Timothy, J.B, Henning, M.A and Swat, H.C On the integity of distance domination in gaphs. Austalian Jounal of Combinatoics, Vol. 10. pp Biogaphical notes Pof. B.K. Ameenal is woking as an Associate Pofesso in the Depatment of Mathematics, Dhanabagiyam Kishnaswamy Mudalia College fo women (Autonomous), Velloe, Tamilnadu state, India. She has been teaching since 1984 and submitted he Ph.D. Thesis at Velloe Institute of Technology (VIT) Univesity in Decembe, He field of specialization is Domination on Discete stuctues in Gaph Theoy. She has published seveal papes in vaious national, intenational confeences and jounals. She has guided 30 students fo thei M.Phil wok. D. R. Selvakuma is Pofesso and Leade of Applied Algeba Division, VIT Univesity, Velloe, Tamilnadu state, India. He is engaged in teaching and eseach activities fo the last 20 yeas. His field of specialization is Algebaic Coding Theoy and Theoetical Compute Science. He has published seveal papes in vaious national, intenational confeences and jounals. He has guided 2 students fo thei Ph.D wok. Received Decembe 2009 Accepted Januay 2010 Final acceptance in evised fom Febuay 2010