To find a non-split strong dominating set of an interval graph using an algorithm

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1 IOSR Journal of Mathematcs (IOSR-JM) e-issn: ,p-ISSN: X, Volume 6, Issue 2 (Mar - Apr 201), PP To fnd a non-splt rong domnatng set of an nterval graph usng an algorthm Dr A Sudhakaraah*, A Sreenvasulu 1,V Rama Latha 2, E Gnana Deepka, Department of Mathematcs, S V Unversty, Trupat , Andhra Pradesh, Inda Abract: In graph theory, a connected component of an undrected graph s a sub graph n whch any two vertces are connected to each other by paths For a graph G, f the subgraph of G tself s a connected component then the graph s called connected, else the graph G s called dsconnected and each connected component sub graph s called t s components A domnatng set D of graph G=(V,E) s a non-splt rong domnatng set f the nduced sub graph < V-D > s connected The non-splt rong domnaton number of G s the mnmum cardnalty of a non-splt rong domnatng set In ths paper conructed a verfcaton method algorthm for fndng a non-splt rong domnatng set of an nterval graph Keywords: Domnaton number, Interval graph, Strong domnatng set, Strong domnaton number, splt domnatng set 1,2,, n I Introducton Let I = {I 1,I 2,,I n } be the gven nterval famly Each nterval n I s represented by [ a, b ], for Here a s called the left endpont and b the rght endpont of the nterval I Wthout loss of generalty we may assume that all end ponts of the ntervals n I whch are dnct between 1and 2n The ntervals are labelled n the ncreasng order of ther rght endponts Two ntervals and j are sad to ntersect each other, f they have non-empty ntersecton Interval graphs play mportant role n numerous applcatons, many of whch are schedulng problems A graph G ( V, E) s called an nterval graph f there s a one-to-one correspondence between V and I such that two vertces of G are joned by an edge n E f and only f ther correspondng ntervals n I ntersect That s, f [ a, b] and j [ a j, bj], then and j ntersect means ether aj b or a b j Let G be a graph, wth vertex set V and edge set E The open neghbourhood set of a vertex v V s nbd ( v) { u V / uv E} The closed neghbourhood set of a vertex v V s nbd[ v] nbd ( v) { v} A vertex n a graph G domnates tself and ts neghbors A set D V s called domnatng set f every vertex n V D s adjacent to some vertex n D The domnaton uded n [1-2] The domnaton number of G s the mnmum cardnalty of a domnatng set The domnaton number s well-uded parameter We can see ths from the bblography [] on domnaton In [4], Sampathkumar and Pushpa Latha have ntroduced the concept of rong domnaton n graphs Strong domnaton has been uded [5-7] Kull V R et all [8] ntroduced the concept of splt and non-splt domnaton[9] n graphs Also DrA Sudhakaraah et all [10] dscussed an algorthm for fndng a rong domnatng set of an nterval graph usng an algorthm A domnatng set D s called splt domnatng set f the nduced subgraph V D s dsconnected The splt domnaton number of of G s the mnmum cardnalty of a splt domnatng set Let G ( V, E) be a graph and u, v V Then u rongly domnates v f () uv E () deg v deg u s A set D V s a rong domnatng set of G f every vertex n V D s rongly domnated by at lea one vertex n D The rong domnaton number ( ) G of G s the mnmum cardnalty of a rong domnatng set A domnatng set V of a graph G s a Non-splt rong domnatng set f the nduced subgraph D V D s connecteddefne NI ( ) j, f b a j and there do not ex an nterval k such that b ak a If j there s no such j, then defne NI () null N sd () s the set of all neghbors whose degree s greater than degree 5 Page

2 To fnd a non-splt rong domnatng set of an nterval graph usng an algorthm of and also greater than If there s no such neghbor then defnes Nsd ( ) null degree vertex n the set S M (S) s the large hghe II Algorthms 21To fnd a Strong domnatng set (SDS) of an nterval graph usng an algorthm[9] Input : Interval famly I { I1, I2,, I n } Output : Strong domnatng set of an nterval graph of a gven nterval famly Step 1 : S nbd [1] 1 Step 2 : S = The set of vertces n S whch are adjacent to all other vertces n S Step : D The large hghe degree nterval n S Step 4 : LI The large nterval n D Step 5 : If N sd ( LI ) exs Step 51 : a = M(N sd ( LI )) 1 1 Step 52 : b The large hghe degree nterval n nbd a Step 5 : D D { b} goto ep 4 end f else Step 6 : Fnd NI(LI) Step 7 : End Step 61: If NI(LI) null goto ep 7 Step 62 : S nbd[ NI( LI)] 2 Step 6 : S The set of all neghbors of NI ( LI ) whch are greater than or equal to NI ( LI ) Step 64 : S The set of all vertces n S whch are adjacent to all vertces n S 4 Step 65 : c = The large hghe degree nterval n S Step 66 : D D { c} goto ep 4 22To fnd a Non-splt Strong domnatng set (NSSDS) of an nterval graph usng an algorthm Input : Interval famly I= {I 1,I 2,I, I n } Output : Whether a rong domnatng set s a non splt rong domnatng set or not Step1 : S 1 =nbd[1] Step2 : S=The set of vertces n S 1 whch are adjacent to all other vertces n S 1 Step : D =The large hghe degree nterval n S Step4 : LI=The large nterval n D Step5 : If W sd (LI) exs Step 51 : a = M(N sd (LI)) Step 52 : b=the large hghe degree nterval n nbd[a] Step 5 : D = D {b} go to ep 4 End f Else Step 6 : Fnd NI(LI) Step 61 : If NI(LI)=null go to ep 7 Step 62 : S 2 =nbd[ni(li)] Step 6 : S The set of all neghbors of NI ( LI ) whch are greater than or equal to NI ( LI ) Step 64 : S The set of all vertces n S whch are adjacent to all 4 vertces n S Step 65 : c = The large hghe degree nterval n S Step 66 : D D { c} goto ep 4 Step 7 : V={1, 2,, ,n} Step 8 : D =k Step 9 : S N ={V-D }={S 1,S 2,S, ,S k }, k 1 n-k Page

3 Step 10 : for ( = 1 to k 1-1) { For ( j = +1 to k 1 ) { If (S,S j ) E of G then plot S to S j } } The nduced sub graph G 1 =V-D s obtaned Step11 : If W(G 1 )=1 D s non splt rong domnatng set Else D s splt rong domnatng set End To fnd a non-splt rong domnatng set of an nterval graph usng an algorthm III Man Theorems Theroem 1 : Let G be an nterval graph correspondng to an nterval famly I={I 1,I 2,I,------I n } If and j are any two ntervals n I such that D s mnmum rong domnatng set of the gven nterval graph G, j 1 and j s contaned n and f there s at lea one nterval to the left of j that ntersects j and at lea one nterval k to the rght of j that ntersects j then D s a non splt rong domnaton Proof : Let G be an nterval graph correspondng to an nterval famly I = {I 1,I 2,I,------I n } Let and j be any two ntervals n I such that D,where D s a mnmum rong domnatng set of the gven nterval graph G, j 1 and j s contaned n and suppose there s at lea one nterval to the left of j that ntersects j and at lea one nterval k to the rght of j that ntersects jthen t s obvously we know that j s adjacent to k n the nduced subgraph <V-D >Then there wll be a connecton n <V-D > to ts left Interval famly I As follows an algorthm wth lluraton for neghbours as gven nterval famly I We conruct an nterval graph G from nterval famly I={1,2,, } as follows nbd[1]={1,2,}, nbd[2]={1,2,,4}, nbd[]={1,2,,4,6}, nbd [4]={2,,4,5,6}, nbd [5]={4,5,6,7}, nbd[6]={,4,5,6,7,9}, nbd[7]={5,6,7,8,9}, nbd[8]={7,8,9,10}, nbd [9]={6,7,8,9,10}, nbd[10]={8,9,10} N sd (1)={2,}, N sd (2)={,4}, N sd ()={6}, N sd (4)=6, N sd (5)={6}, N sd (6)=null, N sd (7)=null, N sd (8)={9}, N sd (9)=null, N sd (10)=null NI(1)=4, NI(2)=5, NI()=5, NI(4)=7, NI(5)=8, NI(6)=8, NI(7)=10, NI(8)=null, NI(9)=null, NI(10)=null Procedure for fndng a non-splt rong domnatng set of an nterval graph usng an algorthm Step 1: S 1 ={1,2,} Step 2: S={1,2,} Step : D ={} Step 4 : LI= Step 5 : N sd ()={6} Step 51 : a=m(n sd ())=M({6})=6 Step 52 : b=6 Step 5 : D ={} {6}={,6} Step 6 : LI=6 Step 7 : NI(6)=8 Step71: S 2 =nbd[8]={7,8,9,10} Step72: S ={8,9,10} Step7: S 4 ={8,9,10} Step74:c=9 Step75 : D = D {9}={,6} {9}={,6,9} Step 8 : V={1,2,, } 7 Page

4 To fnd a non-splt rong domnatng set of an nterval graph usng an algorthm Step 9 : D = Step10 : S N ={1,2,,4,5,6,8,10} Step11 : for =1, j=2, (1,2) E, plot 1 to 2 for = 2, j =, (2,) E, plot 2 to for =, j = 4, (4,5) E, plot 4 to 5 j = 5, (4,6) E, plot4 to 6 for = 4, j = 5, (5,6) E, plot 5 to 6 j = 6, (5,7) E, plot 5 to 7 for = 5, j=6, (6,7) E, plot 6 to7 for = 6, j = 7, (7,8) E, plot 7 to 8 for =7, j = 8, (8,10) E, plot 8 to 10 The nduced sub graph G 1 =<V-D > s obtaned Step12 : W(G 1 )=1 Therefore D s the non splt domnatng set Step1: End Out put : {,6,9} s a non splt rong domnatng set Theorem 2 : If and j are two ntervals n I such that D where D s a mnmum domnatng set of G, j=1 and j s contaned n and f there s one more nterval other than that ntersects j then non-splt rong domnaton occurs n G Proof : Let I = {I 1,I 2,I,I 4,------,I n } be an nterval famly Let j=1 be the nterval contaned n where D, where D s the mnmum rong domnatng set of G Suppose k s an nterval, k and k ntersect j Snce D, the nduced subgraph <V-D > does not contan Further n <V-D >, the vertex j s adjacent to the vertex k and hence there wll not be any dsconnecton n <V-D > Therefore we get non splt rong domnaton n G In ths connecton as follows an algorthm Interval famly I As follows an algorthm wth lluraton for neghbours as gven nterval famly I We conruct an nterval graph G from nterval famly I={1,2,, } as follows nbd[1]={1,2,}, nbd[2]={1,2,,4}, nbd[]={1,2,,4,6}, nbd [4]={2,,4,6,7}, nbd [5]={5,6,7}, nbd[6]={,4,5,6,7,8}, nbd[7]={4,5,6,7,8,9}, nbd[8]={6,7,8,9,10}, nbd [9]={ 7,8,9,10}, nbd[10]={8,9,10} N sd (1)={2,}, N sd (2) ={,4}, N sd () = {4}, N sd (4) ={7}, N sd (5) ={7}, N sd (6)= {7}, N sd (7)= null, N sd (8)= null N sd (9)=null, N sd (10)=null NI(1)=4, NI(2)=5, NI()=5, NI(4)=5, NI(5)=8, NI(6)=9, NI(7)=10, NI(8)=null, NI(9)=null, NI(10)=null 8 9 Procedure for fndng a non-splt rong domnatng set of an nterval graph usng an algorthm Step 1 : S 1 ={1,2,} Step 2 : S={1,2,} Step : D = Step 4 : LI= Step 5 : N sd ()=6 Step 6 : a=6 Step 7 : b=7 Step 8 : D ={} {7}={,7} Step 9 : LI=7 Step10 : NI(7)=10 Step101: S 2 ={8,9,10} Step102 : S ={10} Step10 : S 4 ={10} Step104 : b=10 Step105 : D ={,7,9} Step11 : V={1,2,, } Step12 : D = Step1 : S N ={1,2,4,5,6,8,9} 8 Page

5 To fnd a non-splt rong domnatng set of an nterval graph usng an algorthm Step14 : for =1, j=2,(1,2) E, plot 1 to 2 for = 2, j =, (2,4) E, plot 2 to 4 for =, j = 4, (4,5) E, plot 4 to 5 for = 4, j = 5, (5,6) E, plot 5 to 6 for = 5, j =6, (6,8) E, plot 6 to 8 for = 6,j =7, (8,9) E, plot 8 to 9 The nduced subgraph G 1 = <V-D > s obtaned Step15 : W(G 1 )=1 Therefore D s the non splt rong domnatng set Step16: End Output : {,7,10} s a non splt rong domnatng set Theorem : Let D be a rong domnatng set whch s obtaned by algorthm SDS If, j, k are three consecutve ntervals such that < j< k and j D, ntersects j, j ntersect k and ntere k then non splt rong domnaton occurs n G Proof : Suppose I = {I 1,I 2,I,------I n } be an nterval famly Let, j, k be three consecutve ntervals satsfyng the hypothess Now and k ntersect mples that and k are adjacent nduced sub graph < V\D > an algorthm as follows Interval famly I As follows an algorthm wth lluraton for neghbours as gven nterval famly I We conruct an nterval graph G from nterval famly I={1,2,, } as follows nbd[1]={1,2,}, nbd[2]={1,2,,4}, nbd[]={1,2,,4,5}, nbd [4]={2,,4,5,6}, nbd [5]={,4,5,6,7}, nbd[6]={4,5,6,7,8}, nbd[7]={5,6,7,8,9}, nbd[8]={6,7,8,9}, nbd [9]={7,8,9,10}, nbd[10]={9,10} N sd (1)={2,}, N sd (2)={,4}, N sd ()=null, N sd (4)=null, N sd (5)=null, N sd (6)=null, N sd (7)=null, N sd (8)=null, N sd (9)=null, N sd (10)=null NI(1) = 4, NI(2) = 5, NI() = 6, NI(4) = 7, NI(5) = 8, NI(6) = 9, NI(7)=10, NI(8)=10, NI(9) = null, NI(10) = null Procedure for fndng a non-splt rong domnatng set of an nterval graph usng an algorthm Step 1 : S 1 ={1,2,} Step 2 : S={1,2,} Step : D = Step 4 : LI= Step 5 : NI()=6 Step 6 : Nbd[6]={4,5,6,7,8} Step 61: S = {6,7,8} Step 62 : S = {6,7,8} Step 6 : S 4 = {6,7,8} Step 64 : c=8 Step 65 : D ={,8} Step 7 : LI=8 Step 8 : NI(8)=null Step 9 : V={1,2,, } Step10 : D =2 Step11: S N ={1,2,4,5,6, 9,10} Step12 : for =1, j=2,(1,2) E, plot 1 to 2 for =2, j=, (2,4) E, plot 2 to 4 for =, j=4, (4,5) E, plot 4 to 5 j=5, (4,6) E, plot 4 to 6 for =4, j=5, (5,6) E, plot 5 to 6 j=6, (5,7) E, plot 5 to 7 9 Page

6 To fnd a non-splt rong domnatng set of an nterval graph usng an algorthm for =5, j=6, (6,7) E, plot 6 to 7 for =6,j=7, (7,9) E, plot 7 to 9 The nduced sub graph G 1 s obtaned Step1:W(G 1 )=1 Therefore D s the non splt rong domnatng set Step14: End Output: {,8} s a non splt rong domnatng set IV Conclusons We uded the non-splt rong domnaton n nterval graphs In ths paper we dscussed a verfcaton method algorthm for fndng a non-splt rong domnatng set of an nterval graph Acknowledgements The authors would lke to express ther grattude of the anonymous referees for ther suggeons and nsprng comments on ths paper References [1] T W Haynes, ST Hedetnem and PJSlater, Fundamentals of domnaton n graphs, Marcel Dekker, Inc, New York (1998) [2] T W Haynes, ST Hedetnem and PJSlater, Domnaton n Graphs: advanced topcs, Marcel Dekker, Inc, New York (1998) [] Hedetnem, S T, Laskar, R C, Bblography on domnaton n graphs and some basc defntons of domnaton parameters, Dscrete Mathematcs 86 (1 ), (1990), [4] Sampathkumar, E, and Pushpa Latha, Strong weak domnaton and domnaton balance n graph, Dscrete Math Vol 161, 1996, p [5] Domke etal, On parameters related to rong and weak domnaton n graphs, Dscrete Math Vol 258, (2002) p,1-11 [6] Hattngh JH, Hennng MA, On rong domnaton n Graphs, J Combn Math Combn Comput Vol26, (1998) p 7-92 [7] Rautenbach, D, Bounds on the rong domnaton number, Dscrete Math Vol 215, (2000) p [8] Kull, V R and Janakram, B, The Non-Splt domnaton number of graph, Internatonal Journal of management and syems Vol19 No2, p , 2000 [9] Maheswar, B, Nagamun Reddy, L, and A Sudhakaraah, Splt and Non-splt domnatng set of crcular-arc graphs, J curr Sc Vol No2, (200)p [10] Dr A Sudhakaraah, V Rama Latha, E Gnana Deepka and TVenkateswarulu, To Fnd Strong Domnatng Set and Splt Strong Domnatng Set of an Interval Graph Usng an Algorthm, IJSER, Vol 2,(2012), Page