Non-Split Restrained Dominating Set of an Interval Graph Using an Algorithm

Transcription

1 Internatonal Journal of Advancements n Research & Technology, Volume, Issue, July- ISS - on-splt Restraned Domnatng Set of an Interval Graph Usng an Algorthm ABSTRACT Dr.A.Sudhakaraah *, E. Gnana Deepka, V. Rama Latha, T.Venkateswarulu Department of Mathematcs, S. V. Unversty, Trupat-, Andhra Pradesh, Inda. E-mal:sudhamath.svu@gmal.com *Correspondng Author Among the varous applcatons of the theory of Restraned domnaton, the most often dscussed s communcaton network. There has been persstent n the Algorthmc aspects of nterval graphs n past decades spurred much by ther numerous applcaton of an nterval graphs correspondng to an nterval famly I. A set D V ( G ) s a Restraned domnatng set of a graph G, f every vertex not n D s adjacent to a vertex n D and to a vertex n V D. In graph theory, a connected component of an undrected graph s a subgraph n whch any two vertces are connected to each other by paths. For a graph G, f the nduced subgraph of G tself s a connected component then the graph G s called connected. A Restraned domnatng set RDS of a graph G( V, E ) s a on-splt restraned domnatng set, f the nduced subgraph V RDS s connected. In ths paper we ntroduce an Algorthm to fnd a on-splt Restraned domnatng set of an nterval graph. Key words: Interval famly, nterval graph, connected graph, restraned domnatng set, on-splt restraned domnatng set.. ITRODUCTIO The research of the domnaton n graphs has been an evergreen of the graph theory. Its basc concept s the domnatng set and the domnaton number. The theory of domnaton n graphs was ntroduced by Ore [] and Berge []. A survey on results and applcatons of domnatng sets was presented by E.J.Cockayane and S.T.Hedetnem []. In Kull et.al ntroduced the concept of on-splt domnaton [] and studed these parameters for varous standard graphs and obtaned the bounds for these parameters. In general an undrected graph G ( V, E) s an nterval graph(ig), f the vertex set V can be put nto one-toone correspondence wth a set of ntervals I on the real lne R, such that two vertces are adjacent n G, f and only f ther correspondng ntervals have non-empty ntersecton. The set I s called an nterval representaton of G and G s referred to as the ntersecton graph I. Let I I, I, I, I,... I n be any nterval famly where, each I s an nterval on the real lne and I a, b Copyrght ScResPub. for,,,,... n. Here a s called the left end pont labelng and b s the rght end pont labelng of loss of generalty we assume that all end ponts of the I.Wthout ntervals n I are dstnct numbers between and n. Two ntervals and j are sad to be ntersect each other f they have non empty ntersecton. Also we say that the ntervals contans both ts end ponts and that no two ntervals share a common end pont. The ntervals and vertces of an nterval graph are one and the same thng. The graph G s connected, and the lst of sorted end pont s gven and the ntervals n I are ndexed by ncreasng rght end ponts, that sb b b... b n. Let G ( V, E ) be a graph. A set D V ( G ) s a domnatng set of G f every vertex n V / Ds adjacent to some vertex n D. A set S V s a restraned domnatng set (RDS) f every vertex not n S s adjacent to a vertex n S and to a vertex n V S. Every graph has a RDS, snce S V s such a set. The Restraned domnaton number of G, denoted by r ( G ), s the mnmum cardnalty of a RDS of G. A RDS S s called a r ( G) -set of G f S r ( G ).

2 Internatonal Journal of Advancements n Research & Technology, Volume, Issue, July- ISS - The concept of Restraned domnaton was ntroduced by Telle and Proskurowsk [], albet ndrectly, as a vertex parttonng problem. One applcaton of domnaton s that of prsoners and guards. For securty, each prsoner must be seen by some guard; the concept s that of domnaton. However, n order to protect the rghts of prsoners, we may also requre that each prsoner s seen by another prsoner; the concept s that of restraned domnaton. A Restraned domnatng set RDS of G s connected Restraned domnatng set, f the nduced subgraph V RDS s connected..e., A Restraned domnatng set RDS of a graph G( V, E ) s a on-splt Restraned domnatng set, f the nduced subgraph V RDS s connected. In ths connecton ntroduce the Restraned domnatng set usng an Algorthm [,,,]. For fndng the Restraned domnaton [], through an algorthm, we consder a connected Interval graph. In ths Connected Interval graph the vertces are ordered by IG orderng. Frst of all we treat none of a vertex of VG ( ) s a member of Restraned domnatng set RDS. Then nsert vertces one by one by tastng ther consstency. If a vertex v s domnated by at least two vertces then leave t, otherwse take the hghest numbered adjacent vertex from vas [] a member of RDS f t s not adjacent to the next member of vor [] v s not the last vertex. Let us assocate a new term M () v for a vertex v V, for all,,,..., k( k ( v ) ) to each adjacent vertces of v n order to IG orderng of ntervals n the followng way: Wth M ( v) max ( v ) M ( v) max [ v] M ( v ) In connecton wth the hghest adjacent vertex of v, we call ths M () v as the p - th numbered adjacent vertex of v. Let u, v V. If for some (,,,..., ( v) ), ( v) p such that u M () v, then u s called the p - th numbered adjacent vertex of v. The purpose of ths paper s to fnd the on-splt Restraned domnatng set of an Interval graph. MAI THEOREMS j. THEOREM : Let I,,..., } be an n Interval famly { n and G s an Interval graph correspondng to I. If and j are any two ntervals n I such that RDS,where RDS s a Restraned Domnatng Set, j and j s contaned n, f there s at least one nterval to the left of j that ntersect j and there s at least one nterval k to the rght of j that ntersect j. Then the Restraned domnaton occurs n G and the non-splt restraned domnatng set V RDS s connected as RDS. Proof : Let I {,,..., } be the gven n Interval famly n and G s an nterval graph correspondng to I. Frst we wll fnd the Restraned domnatng set correspondng to G. Suppose there s at least one nterval k to the rght of j that ntersect j. Then t s obvous that j s adjacent to k n V RDS, so that there wll not be any dsconnecton n V RDS. Snce, there s at least one nterval to the left of j that ntersect j, there wll not be any dsconnecton n V RDS, to ts left. Thus we get on-splt Restraned domnaton n G. In ths procedure we also fnd Restraned domnatng set of an nterval graph towards an algorthm wth an llustraton as follows,. A ALGORITHM FOR RESTAIED DOMIATIG SET OF A ITERVAL GRAPH Input: An Interval graph G ( V, E) wth IG orderng vertex set V {,,..., n}. Output: Restraned Domnatng Set RDS Step : Set f ( j), j,,..., n; Step : Set, D ; Step.: Compute W ( f ) Step.: If W ( f ) then v [ ] Set f M ( )), f ( M ( )) ; ( take RDS = { M ( )}. Step.: else f ( f ), f ( v) W s not the last vertex, then Step..: f f ( M ( )), M ( ) s adjacent to M ( ) RDS remans unchanged. end f; Step..: otherwse f M ( ) s not adjacent to M ( ) f ( M ( )), Copyrght ScResPub.

3 Internatonal Journal of Advancements n Research & Technology, Volume, Issue, July- ISS - Set f ( M ( )) take RDS = RDS { M ( )} end f; else f ( f ), s the last vertex, then W RDS remans unchanged. end f; Step.: else f ( f ), then W RDS remans unchanged. end f; Step.: Calculate and go to Step. and contnue untl the last vertex. end RDS. ow we wll fnd the Restraned domnatng set of an nterval graph wth an llustraton usng the above algorthm as follows, Fg.: Interval famly I nbd [] = {,,}, nbd [] = {,,,}, nbd [] = {,,,,}, nbd [] = {,,,,}, nbd [] = {,,,}, nbd [] = {,,,,,}, nbd [] = {,,,,}, nbd [] = {,,,}, nbd [] = {,,,,}, nbd [] = {,,} To fnd the Restraned Domnatng Set, we have to compute all p - th numbered adjacent vertces. TABLE M v \ v M v M v M v M v - - M v M v Frst set f ( j), j V. In Step, set, RDS =, that s ntally RDS s empty. Step repeats for n tmes. Here n, the number of vertces n the nterval graph G. table. We follow the teratons of an llustraton through the Iteraton (): For the frst teraton,, w f f ( []) w f f f f () The frst condton of f-end f s satsfed.snce w f, we fnd M, M Then set f, f Also set RDS = { } RDS = { } Iteraton (): For the second teraton,,,, w f f ( []) w f f f f f So, n ths teraton RDS could not be calculated. Hence RDS remans same and s beng ncreased to. Iteraton (): For the thrd teraton,,,, w f f ( []) w f f f f f () f () In ths teraton RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the fourth teraton,,,, w f f ( []) w f f f f f f () In ths teraton RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the ffth teraton,,, Copyrght ScResPub.

4 Internatonal Journal of Advancements n Research & Technology, Volume, Issue, July- ISS - W f f ( []) W f f f f f The frst condton of f-end f s satsfed. Snce w f, we fnd M, M M ), () ( M We fnd Then set f, f Also set RDS = RDS { } RDS = {,} The teraton number s beng ncreased to. Iteraton (): For the sxth teraton,,,,, W ( f ) f ( []) W ( f ) f f f f f f In ths teraton RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the seventh teraton,,,, W f f ( []) W f f f f f f In ths teraton RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the eghth teraton,,, w f f ( []) w f f f f () f () Here the Restraned domnaton crtera s not satsfed. The else-f condton of f-end f s satsfed. ow f ( M()) f and M () M (). So RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the nnth teraton s adjacent to,,,, w f f ( []) w f f f f f f () In ths teraton RDS could not be calculated. Hence RDS remans unchanged and s beng ncreased to. Iteraton (): For the tenth teraton,, W f f ( []) W f f f f The frst condton of f-end f s satsfed. Snce w f, M, M we fnd Then set f, f Also set RDS = RDS { } RDS = {,} {} {,,} RDS = {,,} RDS = The cardnalty of RDS =. Thus we get the on-splt Restraned domnatng set V RDS as follows, Fg.: Vertex nduced subgraph V RDS - Connected graph from G. THEOREM : Let G be an Interval graph correspondng to an n Interval famly I,,..., }. If and j are any { n two ntervals n I such that RDS, j, j ntersects and f there s one more nterval that ntersects j or contans j. Then Restraned domnaton occurs n G and the non-splt restraned domnatng set V RDS s connected as RDS. Copyrght ScResPub.

5 Internatonal Journal of Advancements n Research & Technology, Volume, Issue, July- ISS - Proof : Let I {,,..., } be the gven n Interval famly Copyrght ScResPub. n and G s an nterval graph correspondng to I. Frst we wll fnd the Restraned domnatng set correspondng to G. ow let j be an nterval contaned n an nterval k or ntersects k whch s not n the Restraned domnatng set. Suppose j ntersects, snce RDS, V RDS does not contan. Further n V RDS, the vertex j s adjacent to the vertex k, snce j s contaned n k or j ntersects k and hence there wll not be any dsconnecton n V RDS. Therefore we get on-splt domnatng n G. ext we wll fnd the Restraned domnatng set as follows from an nterval famly usng Algorthm as explaned n secton.. Fg.: Interval Famly I nbd [] = {,,}, nbd [] = {,,,}, nbd [] = {,,,,}, nbd [] = {,,,,}, nbd [] = {,,,,}, nbd [] = {,,,}, nbd [] = {,,,,,}, nbd [] = {,,,}, nbd [] = {,,} To fnd the Restraned Domnatng Set, we have to compute all p - th numbered adjacent vertces. TABLE M v \ v M v M v M v M v - - M v M v Frst set f ( j), j V. In Step, set,, that s ntally RDS s empty. Step repeats for n RDS = tmes. Here n, the number of vertces n the nterval graph G. As follows the teratons through the table, Iteraton () : For the frst teraton w f f ( []),, w f f f f The frst condton of f-end f s satsfed. Snce w f, we fnd M, M Then set f, f Also set RDS = { } RDS = {} Iteraton (): For the second teraton,,,, w f f ( []) w f f f f f So, n ths teraton RDS could not be calculated. Hence RDS remans same and s beng ncreased to. Iteraton () For the thrd teraton,,,, w f f ( []) w f f f f f () f () In ths teraton RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the fourth teraton,,,, w f f ( []) w f f f f f f () In ths teraton RDS remans unchanged. The teraton number s beng ncreased to. Iteraton ():

6 Internatonal Journal of Advancements n Research & Technology, Volume, Issue, July- ISS - For the ffth teraton,,,, W f f ( []) W f f f f f f Here the Restraned domnaton crtera s not satsfed. The else-f condton of f-end f s satsfed. ow f ( M()) f and M () s adjacent to M (). So RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the sxth teraton,,, W ( f ) f ( []) W ( f ) f f f f The frst condton of f-end f s satsfed. Snce w f, M, M we fnd Also set Then set f, f RDS = RDS {} RDS = {} {} {,} The teraton number s beng ncreased to. Iteraton (): For the seventh teraton,,,,, W f f ( []) W f f f f f f f In ths teraton RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the eghth teraton,,, w f f ( []) w f f f f f () In ths teraton RDS could not be calculated. The teraton number s beng ncreased to. Copyrght ScResPub. Iteraton (): For the nnth teraton,, w f f ( []) w f f f f In ths teraton RDS could not be calculated. Hence RDS remans unchanged. RDS = {,} RDS = The cardnalty of RDS =. Thus we get the on-splt restraned domnatng set V RDS as follows, Fg.: Vertex nduced subgraph V RDS - Connected graph from G. THEOREM : Let us consder an n nterval famly I {,,..., n } and G be an nterval graph of I. If, j, k are three consecutve ntervals such that j k and j RDS,, ntersects j, j ntersect k and ntersect k. Then Restraned domnaton occurs n G and the non-splt restraned domnatng set V RDS s connected as RDS. Proof : Let I {,,..., n } be an n nterval famly and G be an nterval graph of I. Let, j, k be three consecutve ntervals satsfy the hypothess. ow and k ntersect mples that and k are adjacent n V RDS. So that there wll not be any dsconnecton n V RDS. ow we wll fnd Restraned domnatng set usng Algorthm as gven n secton. as follows. For ths consder the followng nterval famly,

7 Internatonal Journal of Advancements n Research & Technology, Volume, Issue, July- ISS - Fg.: Interval Famly I nbd [] = {,,}, nbd [] = {,,,}, nbd [] = {,,,,}, nbd [] = {,,,,}, nbd [] = {,,,,}, nbd [] = {,,,,}, nbd [] = {,,,,}, nbd [] = {,,,,}, nbd [] = {,,,}, nbd [] = {,,}. To fnd the Restraned Domnatng Set, we have to compute all p - th numbered adjacent vertces. TABLE M v \ v M v M v M v M v - - M v Frst set f ( j), j V. In Step, set, RDS =, that s ntally RDS s empty. Step repeats for n tmes. n the number of vertces n the nterval graph G. Here, As follows teratons, Iteraton () : For the frst teraton,, w f f ( []) Copyrght ScResPub. w f f f f () The frst condton of f-end f s satsfed. Snce w f, we fnd M, M Then set f, f Also set RDS = { } RDS = { } Iteraton (): For the second teraton,,,, w f f ( []) w f f f f f So, n ths teraton RDS could not be calculated. Hence RDS remans same and s beng ncreased to. Iteraton () For the thrd teraton,,,, w f f ( []) w f f f f f () f () In ths teraton RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the fourth teraton,,,, w f f ( []) w f f f f f f () In ths teraton RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the ffth teraton,,,, W f f ( []) W f f f f f f Here the Restraned domnaton crtera s not satsfed. The else-f condton of f-end f s satsfed. f ( M ()) f and M () ow s adjacent to M (). So RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the sxth teraton,,,, W ( f ) f ( []) W ( f ) f f f f f The frst condton of f-end f s satsfed. Snce w f, we fnd M, M Then set f, f Also set RDS = RDS {} RDS = {} {} {,} The teraton number s beng ncreased to. Iteraton ():

8 Internatonal Journal of Advancements n Research & Technology, Volume, Issue, July- ISS - For the seventh teraton,,,, W f f ( []) W f f f f f f In ths teraton RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the eghth teraton,,,, w f f ( []) w f f () f f f () In ths teraton RDS remans unchanged. The teraton number s beng ncreased to. Iteraton (): For the nnth teraton,,, w f f ( []) w f f f f f () COCLUSIO We study the on-splt restraned domnatng set problem on an nterval graph correspondng to an nterval famly I. Gven an nterval model wth end ponts sorted. We presented an algorthm to solve the Restraned domnaton problem on nterval graphs. We extended the results to solve the on- splt restraned domnaton problem on nterval graphs usng an algorthm. ACKOWLEDGMET The Authors are greatful to the referees for ther valuable comments whch have lead to mprovements n the presentaton of the paper. Ths research was supported n part by the S.V.Unversty, Trupat, IDIA. REFERECES [] O.Ore, Theory of Graph, Amer, Math.Soc.Colloq.Publ., Provdence (), P.. [] C.Berge, Graphs and Hyperactve graphs,orth Holland, Amsterdram n graphs, etworks, Vol.(), -. In ths teraton RDS could not be calculated. Hence RDS remans unchanged and s beng ncreased to. Iteraton (): For the tenth teraton,, W f f ( []) W f f f f W f s the last vertex, then RDS remans unchanged. RDS = {,}, RDS = The cardnalty of RDS =. Thus we get the on-splt Restraned domnatng set V RDS as follows, Fg.: Vertex nduced subgraph V RDS - Connected graph from G [] E.J.Cockayne, S.T.Hedetnem, Towards a theory of domnaton n graphs, etworks, Vol.(), -. [] V.R.Kull, B.Janakram, The on-splt domnaton number of a graph, Indan J.Pure.Appled Mathematcs, Vol.(), -, May. [] J.A.Telle and A.Proskurowsk, Algorthms for vertex portonng problems on partal k-trees. Sam J.Dscrete Math. ) -. [] M.Pal, S.Mondal, D.Bera and T.K. Pal, An optmal parallel algorthm for computng cutvertces and blocks on nterval graphs, Internatonal Journal of Computer Mathematcs, () -. [] M.C. Golumbc, Algorthmc Graph Theory and Perfect Graphs, Academc Press, ew York,. [] Tarasankar Pramank, Sukumar Mondal and Madhumangal Pal, Mnmum -tuple domnatng set of an nterval graph. [] Dr.A.Sudhakaraah, E.Gnana Deepka, V.Rama Latha, To fnd a -tuple domnatng set of an nduced subgraph of a non-splt domnatng set of an nterval graph usng an algorthm, Internatonal Journal of Engneerng Research and Technology, ISS:-, Vol. Issue, May-. [] G.S.Domke, J.H. Hattngh, M.A.Hennng, and L.R.Markus, Restraned domnaton n graphs wth mnmum degree two. J.Combn.Math.Combn.Comput. () -. Copyrght ScResPub.

9 Internatonal Journal of Advancements n Research & Technology, Volume, Issue, July- ISS - Copyrght ScResPub.