Dr. A. Sudhakaraiah* V. Rama Latha E.Gnana Deepika

Transcription

1 Internatonal Journal Of Scentfc & Engneerng Research, Volume, Issue 6, June-0 ISSN - Splt Domnatng Set of an Interval Graph Usng an Algorthm. Dr. A. Sudhakaraah* V. Rama Latha E.Gnana Deepka Abstract : We study the problem of computng mnmum domnatng sets of n ntervals on lnes. Interval graphs are rch n combnatoral structures and have found applcatons n several dscplnes such as traffc control, ecology, bology, computer scences and pa rtcularly useful n cyclc schedulng and computers storage allocaton problems etc. In ths paper we dscussed the notons new algorthms for splt domnaton n graphs usng (mnmum domnatng set) MDS algorthm. We get many bounds and splt domnaton number. Key Words : Interval famly, Interval graph, Connected graph, Domnatng Set, splt domnatng set, Connected domnatng set, splt domnatng number. INTRODUCTION W e consder the problem of ncrementally computng a mnmal domnatng set of an nterval graph after the nserton or deleton of a set of lnes. Let I { I, I,..., I } n be the gven nterval famly. Each nterval n I s represented by [ a, b ], for =,,..., n. 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 dstnct between and n. 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. 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, b ], then and j ntersect means ether aj < b are a < bj. Kull.V.R et.all [] ntroduced the concept of splt and non splt domnaton n graphs and also n Maheswar, B et all[]. A non empty set D V of a graph G s a domnatng set[] of G f every vertex n V D s adjacent to some vertex n D. The domnaton number (G) s the mnmum cardnalty taken over all the mnmal domnatng sets of G. A domnatng set Correspondng Author : Dr.A.Sudhakaraah, Asst. Professor,Dept. of Mathematcs,S.V.Unversty,Trupat-0,Andhra Pradesh,Inda.E_mal : sudhamath.svu@gmal.com. V.Rama Latha,Research Scholar, Dept. of Mathematcs, S.V.Unversty,Trupat-0,Andhra Pradesh,Inda.E_mal : v.ramalatha@gmal.com E.Gnana Deepka, Dept. of Mathematcs, S.V.Unversty,Trupat- 0,Andhra Pradesh,Inda. j j s sad to be mnmum domnatng set f t s domnaton number s mnmum. A graph G s sad to be connected, f there s a path between any two vertces of G. Otherwse t s dsconnected.. Let G be a connected graph. If V s a vertex set of G such that G -V s dsconnected then the vertex v s called a cut vertex. A domnatng set D of G s a connected domnatng set [4] f the nduced sub graph D s connected. A domnatng set D set (SDS) f the nduced subgraph V V of a graph G s a splt domnatng D s dsconnected. The splt domnaton number s the mnmum cardnalty of a splt domnatng set. It s denoted by s( G ). The neghbourhood of a vertex v V s set consstng all vertces adjacent to v (ncludng v). It s denoted by nbd[] v.let nbd[] be defned as the set of vertces adjacent to ncludng. Let max() denotes the largest nterval n nbd[]. Guruprakash. C. D.,Mallkarjuna Swamy.B. P [], Mnmum Matchng Domnatng Sets and ts Apllcatons n Wreless Networks defne Next() = j f and only f b a jand there does not exst an nterval k such that b < a < a k j. If there s no such j, we defne Next ALGORITHM OF MDS Input : Let I {,,..., n} be Interval famly Output: Mnmum domnatng set of an nterval Step : Let MDS = {max()}. Step : LI Step : Compute Next LI. The largest nterval n MDS. Step 4 : If Next LI null the goto step. Step : Fnd max Next LI. Step 6 : If max(next(li)) does not exst then max(next(li)) Next(LI). IJSER 0

2 Internatonal Journal Of Scentfc & Engneerng Research, Volume, Issue 6, June-0 ISSN - that ntersects j and there s no nterval k to the rght of j Step : MDS MDS max Next LI goto step. Step : End. AN ALGORITHM FOR FINDING SPLIT DOMINATING SET. Input: Interval famly I {,,,..., n } Output: Splt domnatng set and nduced sub graph of an nterval graph G s dsconnected. Step : Set V U and U where 0 to n. Step : Fnd MDS D for some where 0 n. Step : = MDS. Step 4 : V MDS S, where 0 to n Step : for 0 to n { If max S n then p max S[ ] else If max S n- then p max S[ ] ( p values depends on ) else p max S[ ] for j { If S,S j to p E n G Plot a lne from S[] to S[j] } } The nduced sub graph G obtaned. Step 6 : f w G w G G s dsconnected then go to step Step : S max S 0 n G. Step : LI The largest nterval n S. Step : Compute Next LI n G. Step 0 : If Next LI null the go to step 4. Step : Fnd max Next LI n G Step : If max (Next(LI )) does not exst then max (Next(LI )) Next(LI ) n G. Step :S {S } {max (Next(LI ))} go to step. Step 4 : End. 4 MAIN THEOREMS Theorem 4. : Let I { I, I,..., I n } be an nterval famly and G s an nterval graph correspondng to I. If and j are any two ntervals n I such that D, j and j s contaned n and f there s at least one nterval to the left of j that ntersects j, then splt domnaton occurs n G. Proof: Let I { I, I,..., I n } be the gven nterval famly and G t s correspondng nterval graph. Let and j be any two ntervals n I such that j s contaned n. Let D be a domnatng set of G and f D. Let m be an nterval n I whch s to the left of j and ntersect j. Further by our assumpton there s no nterval k > j such that k ntersect j. Now j s contaned n mples j <. Therefore j s not adjacent to any vertex n the set {,,,..., n} and D mples that the nduced subgraph V D does not contan. Ths mples that there s a dsconnecton at j. Hence we have splt domnaton n G. Our am to show that an algorthm to fnd splt domnatng set of an nterval graph wth an llustraton. ILLUSTRATTION We construct an nterval graph G from an nterval famly I {,,,...,0} as follows. nbd,,4, nbd,,,4, nbd,,4, nbd 4,,,4,, nbd 4,,6,, nbd 6,6,,,0, nbd,6,,,0, nbd 6,,,,0, nbd,,0, nbd 0 6,,,,0. max 4, max 4, max 4, max 4, max, max 6 0, max 0, max 0, max 0, max 0 0. Next(), Next(), Next(), Next(4) 6, Next(), Next(6), Next(), Next() null, Next() null, Next(0) Input: Interval famly gven n fg. Step : MDS 4 Step : LI 4. Step : Next(4) = 6. Step 4 : max(6) 0. Step : MDS {4} {0} 4,0 goto step. Step 6: LI 0. Step : Next(0) 4 Fgure. Interval famly 6 0 IJSER 0

3 Internatonal Journal Of Scentfc & Engneerng Research, Volume, Issue 6, June-0 ISSN - Step : End. Out put : {4,0} s the mnmum domnatng set of an nterval Step : V {,,...0}. Step : MDS {4,0}. Step :. Step 4 :S {,,,,6,,,}. Step 4: for 0 then max S[0] 4 p 4. j to. ( S[0] S []) (,) E n G, plot a lne from to ( S[0], S[]) (,) E n G, ( S[0], S[]), E n G for, max S[] 4 p 4. j to. ( S[], S []) (,) E n G, plot a lne from to. ( S[], S []) (,) E n G. for, max S[] 4 p 4. j to. ( S[], S []) = (,) E n G. for, max S[] p 6 j 4 to 6 ( S[], S [4]) (,6) E n G, plot a lne from to 6. ( S[], S []) (,) E n G, plot a lne from to. ( S[], S [6]) (,) E n G for 4, max S 4 0 p 0 j to ( S[4], S []) = (6,) E n G, plot a lne from 6 to. ( S[4], S [6]) (6,) E n G, plot a lne from 6 to. ( S[4], S []) (6,) E n G. for,max S[] 0 p 0. j 6 to. ( S[], S [6]) (,) E n G, plot a lne from to. ( S[], S []) =(,) E n G. for 6, max S[6] 0 p 0. j to. ( S[6], S []) (,) E n G, plot a lne from to. The nduced sub graph G = G-MDS s obtaned. Step 6 : w G w G Therefore G s dsconnected, then go to step Step : S max S[0]. Step : LI. Step : Next(). Step 0: m ax(). Step :S {} {} {,} goto step. Step : LI. Step : Next(). Step4: max() Step :S {,} {} = {,,} goto step. Step 6: LI. Step : Next() Step : End. Out put : {4,0} s the splt domnatng set of the nterval famly as n fg.,domnatng set of G s {,,} and s < (G) Theorem 4. : Let D be a mnmum domnatng set of the gven nterval If and j are any two ntervals n I such that j s contaned n and f there s no other nterval k that ntersects j the splt domnaton occurs n G. {,,..., } Proof: Let I I I I n be an nterval famly and G s an nterval graph correspondng to I. Let and j be any two ntervals n I such that j s contaned. If there s no nterval k that ntersect j. Then clearly les n the domnatng set D. Further n nduced sub graph V D the vertex j s not adjacent to any other vertex and nfact j becomes as solated vertex n nduced sub graph V D. Hence we get splt domnaton. ILLUSTRATION As follows an algorthm wth an llustraton for neghbours as gven nterval famly I. We construct an nterval graph G from an nterval famly I {,,...,} as follows nbd[] {, }, nbd[] {,,}, nbd[] {,,4,}, nbd[4] {, 4,, 6}, nbd[] {,4,,6}, nbd[6] {4,, 6,,}, nbd[] {6,,,}, nbd[] {6,,,}, nbd[] {,,} max(), max(), max(), max(4) 6, max() 6, max(6), max(), max(), max(). Next(), Next() 4, Next() 6, Next(4), Next(), Next(6), Next() null, Next() null, Next() Input: Interval famly gven n fg. Step : MDS. Step : LI. Step : Next() = 4. 4 Fgure. Inteval Famly 6 IJSER 0

4 Internatonal Journal Of Scentfc & Engneerng Research, Volume, Issue 6, June-0 4 ISSN - Step 4: max(4) 6. Step : MDS {} {6},6 goto step. Step 6: LI 6. Step : Next(6). Step : max(). Step : MDS {,6} {},6, goto step. Step 0: LI. Step : Next() Step : End. Out put : {,6,} s the mnmum domnatng set of an nterval Input : Let Interval famly I {,,...,}. Smlarly the above method as follows nbd[], max()and Next(). Step : V={,,,4,,6,,,}. Step : MDS Step :. {,6,} by usng above MDS algorthm Step 4:S[ ] {,,4,,,} Step : for 0 then max S[0] p. ( S[0], S []) (,) E n G. j to. for then max S[] 4 p 4 j to 4 ( S[], S []) (,4) E n G,plot a lne from to 4. ( S[], S []) (,) E n G,plot a lne from to. ( S[], S [4]) (,) E n G. for, max S[] 6 p 6. j to. ( S[], S []) (4,) E n G,plot a lne from 4 to ( S[], S [4]) (4,) E n G, ( S[], S []) (4,) E n G. for, max S[] 6 p 6. ( S[], S[4]) (,) E n G, j 4 to. (S[],S[]) (,) E n G. for 4,max S[4] p ' j to. (S[4],S[]) (,) E n G,plot a lne from to. The nduced sub graph G=G-MDS obtaned. Step 6 : w G w G. Therefore G s dsconnected then go to step Step : S max S 0. Step : LI. Step : Next() n G. Step 0: max() n G. Step : S {} {} {,} goto step. Step : LI. Step : Next() n G. Step 4: max() n G. Step :S {{,} {} {,,} goto step. Step 6: LI n G. Step : Next() null n G. Step : End. Output : {,6,} s the splt domnatng set of the nterval famly as n fg.., domnatng set of G s (,,} and s = (G). Theorem 4. : Let I { I, I,..., I n } be an nterval famly and D s a mnmum domnatng set of the gven nterval graph G. If, j, k are any three consecutve ntervals such that j k and f j D, and ntersects j, j ntersect k and does not ntersect k then splt domnaton occurs n G. Proof : Suppose I { I, I,..., I n } be an nterval famly. If, j, k be three consecutve ntervals such that j k and ntersect j, j ntersect k, but does not ntersect k. Suppose j D, where D s a mnmum domnatng set. Then and k are not adjacent n V D. That s there s a dsconnecton between and k provded, there s no m I, m k such that m ntersects k. If such an m exsts, then snce m k we must have m j k. Now m ntersects k mples and j also ntersect. Ths s a contradcton to hypothess. So such a m does not exst hence we get splt domnaton. As usual as follows an algorthm to fnd a splt domnatng set of an nterval 4 6 Fgure. Interval famly We construct an nterval graph from an nterval famly I {,,...,} as follows nbd[] {,}, nbd[] {,,,4}, nbd[] {,,4,}, nbd[4] {,,4,}, nbd[] {,4,,6},nbd[6] {,6,,}, nbd[] {6,,,}, nbd[] {6,,,}, nbd[] {,,}. max(), max() 4, max(), max(4), max() 6, max(6), max(),max(),max(). Next(), Next(), Next() 6, Next(4) 6, Next(), Next(6), Next() null, Next() null, Next() Input: Interval famly gven n fg. Step : MDS. IJSER 0

5 Internatonal Journal Of Scentfc & Engneerng Research, Volume, Issue 6, June-0 ISSN - Step : LI. Step : Next () =. Step 4: max() 6. Step : MDS {} {6},6 goto step. Step 6: LI 6. Step : Next(6). Step : max(). Step : MDS {,6} {},6, goto step. Step 0: LI. Step : Next() Step : End. Out put : {,6,} s the mnmum domnatng set of an nterval Input: Interval famly I {,,..., } as the above method follows nbd[], max()and Next(). Step : V={,,,4,,6,,,} Step : MDS {,6,}. Step :. Step 4: S[ ] {,,4,,,}. Step : for 0 then max S[0] p. j to. ( S[0], S []) (,) E n G. for then max S[] p 4. j to 4. ( S[], S []) (,4) E n G,plot a lne from to 4. ( S[], S []) (,) E n G, plot a lne from to. ( S[], S [4]) (,) E n G. for,max S[] p 4. j to 4. ( S[], S []) (4,) E n G, plot a lne from 4 to. ( S[], S [4]) (4,) E n G. for, max S[] 6 p 6. j 4 to. ( S[], S [4]) (,) E n G, (S[],S[]) (,) E n G. for 4,max S[4] p j to ( S[4], S []) (,) E n G, plot a lne from to. The nduced sub graph G=G-MDS obtaned. Step6 : w G w G Therefore G s dsconnected then go to step. Step : S max S 0. Step : LI. Step : Next() n G. Step 0: max() n G. Step :S {} {} {,} goto step. Step : LI. Step : Next() n G. Step 4: max() n G. Step :S {,} {} {,,} goto step. Step 6: LI n G. Step : Next() null n G. Step : End. Output : {,6,} s the splt domnatng set of the nterval famly as n fg.., domnatng set of G s (,,} and s = (G) and s = (G) CONCLUSIONS In ths paper we dscussed the splt domnatng set of an nterval graph usng an algorthm. We present an algorthm approxmaton scheme for ths problem on an nterval graph correspondng to an nterval famly wth bounded growth. The scheme s robust and Thus returns for any undrected graph gven as nput a meanngful output. For whch an optmal, partal soluton can be obtaned. Whle ths approach s already used for related problems, feasblty when combnng the partal solutons s an ssue for sets that have to reman both domnatng set and splt domnatng set of an nterval graph. We solve ths ssue by a dsconnected processng repar algorthm. REFERENCES [] Kull,V.R. and Janakram B, Ths nonsplt domnaton number of graph. Internatonal Journal of management and systems. Vol.. No., pp. 4-6, 000. [] Maheswar, B., Nagamun Reddy, L., and Sudhakaraah, A.,, Splt and non-splt domnatng set of crcular-arc graphs, J. curr. Sc (),6-, 00. [] Cockayne, E.J. and Hedetmem, S.T., Towards a theory of domnaton n graphs.networks., pp,4-6,. [4] Wel et al, Mnmum connected domnatng sets and maxmal ndepeent sets n unt dsk graphs. Theortcal Computer Scence, -, 006. [] Guruprakash. C. D., Mallkarjuna Swamy. B. P. Mnmum Matchng Domnatng Sets and ts Apllcatons n Wreless Networks. Vol. No.0, -, 0. IJSER 0