On Embedding and NP-Complete Problems of Equitable Labelings

Transcription

1 IOSR Journal o Mathematcs (IOSR-JM) e-issn: , p-issn: X Volume, Issue Ver III (Jan - Feb 5), PP 8-85 wwwosrjournalsorg Ombeddng and NP-Complete Problems o Equtable Labelngs S K Vadya, C M Barasara (Department o Mathematcs, Saurashtra Unversty, Rajkot , Gujarat, INDIA) (Atmya Insttute o Technology and Scence, Rajkot , Gujarat, INDIA) Abstract: The embeddng and NP-complete problems n the context o varous graph labelng schemes were remaned n the ocus o many researchers Here we have explored embeddng and NP-Complete problems or some varants o cordal labelng Keywords: Graph labelng, equtable labelng, embeddng o graphs, NP-complete problems Mathematcs Subject Classcaton (): 5C78, 5C6 I Introducton A graph labelng s an assgnment o ntegers to the vertces or edges, or both, subject to certan condton or condtons Labelng o dscrete structures s a potental area o research and a comprehensve survey on graph labelng can be ound n Gallan [] Denton : For a graph G, a vertex labelng uncton : V( {, } nduces an edge labelng uncton : E( {,} dened as ( e uv) ( u) ( v) Then s called cordal labelng o graph G absolute derence o number o vertces wth label and label s at most and absolute derence o number o edges wth label and label s at most A graph G s called cordal t admts a cordal labelng The concept o cordal labelng was ntroduced by Caht [] n 987 and n the same paper he presented several results on ths newly ntroduced concept Ater ths some labelngs lke prme cordal labelng, A - cordal labelng, product cordal labelng, edge product cordal labelng etc were also ntroduced wth mnor varatons n cordal theme Such labelngs are classed as equtable labelngs Denton : For a graph G, an edge labelng uncton : E( {, } nduces a vertex labelng uncton : V( {,} dened as ( u) { ( uv)/ uv E( } Then s called an edge product cordal labelng o graph G the absolute derence o number o vertces wth label and label s at most and the absolute derence o number o edges wth label and label s also at most A graph G s called edge product cordal graph t admts an edge product cordal labelng In, the concept o edge product cordal labelng was ntroduced by Vadya and Barasara [3] In the same paper they have nvestgated edge product cordal labelng or some standard graphs For any graph G, we ntroduce ollowng notatons: v () = the number o vertces havng label ; v () = the number o vertces havng label ; 3 e () = the number o edges havng label ; 4 e () = the number o edges havng label ; Throughout ths dscusson we consder smple, nte and undrected graph G ( V(, E( ) wth V( p and E( q For all other standard termnology and notaton we reer to Chartrand and Lesnak[4] DOI: 979/ wwwosrjournalsorg 8 Page

2 Ombeddng and NP-Complete Problems o Equtable Labelngs II Embeddng o Equtable graphs Denton : For a graph G, an edge labelng uncton : E( {, } nduces a vertex labelng uncton : V( {,} dened as ( u) { ( uv)/ uv E( } Then s called a total edge product cordal labelng o graph G v e v e () () () () A graph G s called total edge product cordal graph t admts a total edge product cordal labelng The concept o total edge product cordal labelng was ntroduced by Vadya and Barasara [5] Theorem : Any graph G can be embedded as an nduced subgraph o a total edge product cordal graph Proo Label the edges o G n such a way that edges wth label and edges wth label der by at most V be the set o vertces wth label and be the cardnalty o set V and E respectvely Case : When n V n V ( ) ( ) ( ) ( ) (say r ) E be the set o edges wth label whle nv and r The new graph H can be obtaned by addng vertces to the graph G and jon them to arbtrary r member o V Assgn label to these new edges Consequently vertces wll receve label r = n( V ) n( E ) n( V ) n( E ) Thus, v () e () v () e () Case : When n V n V ( ) ( ) ( ) ( ) (say r ) r The new graph H can be obtaned by addng vertces to the graph G and jon them to arbtrary r member o V Assgn label to these new edges Consequently vertces wll receve label r Thus, v () e () v () e () = n( V ) n( E ) n( V ) n( E ) Thus n all the possbltes, the constructed supergraph H satses the condtons or total edge product cordal graph That s, any graph G can be embedded as an nduced subgraph o a total edge product cordal graph Denton 3: : V( {,,, p} be a bjecton For each edge uv, assgn the label ( u) ( v) Functon s called a derence cordal labelng e () e () where e () and e () denote the number o edges labeled wth and not labeled wth respectvely A graph wth a derence cordal labelng s called a derence cordal graph The concept o derence cordal labelng was ntroduced by Ponraj et al [6] n 3 Theorem 4: Any graph G can be embedded as an nduced subgraph o a derence cordal graph Proo G be the gven graph wth p vertces and q edges As per denton o derence cordal labelng assgn labels {,,, p} to the vertces o G E be the set o edges wth label whle be the cardnalty o set Case : When n( E) n( E) (say r ) The new graph H can be obtaned by addng r vertces, say { v, v,, v r } to the graph G and jon them to the vertex wth label Assgn label p to the vertex v or,,, r Consequently r edges wll receve label Hence, e () e () = n( E) n( E) r Case : When n( E) n( E) (say r ) E DOI: 979/ wwwosrjournalsorg 8 Page

3 Ombeddng and NP-Complete Problems o Equtable Labelngs The new graph H can be obtaned by addng a path o r vertces, say { v, v,, v r } to the graph G and jon the vertex v to the vertex wth label p Assgn label p to the vertex v or,,, r Consequently r edges wll receve label Hence, e () e () = n( E) r n( E) Thus n all the possbltes, the constructed supergraph H satses the condtons or derence cordal graph That s, any graph G can be embedded as an nduced subgraph o a derence cordal graph Denton 5: A graph G s called smply sequental there s a bjecton : V( E( {,,, p q} such that or each edge e { uv} E( one has ( e) ( u) ( v) In 983, Bange et al [7] have ntroduced the concept o sequental labelng Denton 6: I a mappng : V( E( {,} s such that or each edge e uv, ( e) ( u) ( v) and the condton v e v e () () () () holds then G s called total sequental cordal graph In, Caht [8] has ntroduced the concept o total sequental cordal labelng as a weaker verson o sequental labelng Theorem 7: Any graph G can be embedded as an nduced subgraph o a total sequental cordal graph Proo Label the vertces o graph G n such a way that vertces wth label and vertces wth label der by at most V be the set o vertces wth label and be the cardnalty o set V and E respectvely Case : When n V n V ( ) ( ) ( ) ( ) (say r ) E be the set o edges wth label whle DOI: 979/ wwwosrjournalsorg 8 Page nv and r The new graph H can be obtaned by addng vertces to the graph G and jon them to arbtrary r member o V Assgn label to these new vertces Consequently edges wll receve label r = n( V ) n( E ) n( V ) n( E ) Thus, v () e () v () e () Case : When n V n V ( ) ( ) ( ) ( ) (say r ) r The new graph H can be obtaned by addng vertces to the graph G and jon them to arbtrary r member o V Assgn label to these new vertces Consequently edges wll receve label r Thus, v () e () v () e () = n( V ) n( E ) n( V ) n( E ) Thus n all the possbltes, the constructed supergraph H satses the condtons or total sequental cordal graph That s, any graph G can be embedded as an nduced subgraph o a total sequental cordal graph Denton 8: : V( {,,, p} be a bjecton For each edge uv, assgn the label ether ( u) ( v ) or ( v) ( u ) and the label otherwse s called a dvsor cordal labelng e () e () A graph wth a dvsor cordal labelng s called a dvsor cordal graph In, Varatharajan et al [9] have ntroduced the concept o dvsor cordal labelng Theorem 9: Any graph G can be embedded as an nduced subgraph o a dvsor cordal graph Proo G be the gven graph wth p vertces and q edges As per denton o dvsor cordal labelng assgn labels {,,, p } to the vertces o G E be the set o edges wth label whle Case : When ne ne (say r ) be the cardnalty o set E

4 Ombeddng and NP-Complete Problems o Equtable Labelngs The new graph H can be obtaned by addng r vertces to the graph G, say v, v,, v r and jon them to vertex wth label Assgn label p to the vertex v or,,, r Consequently r edges wll receve label Hence, e () e () Case : When ne ne = (say r ) r The new graph H can be obtaned by addng a path o r vertces, say v, v,, v r to the graph G and jon the vertex v to the vertex wth label p Assgn label p to the vertex v or,,, r Consequently r edges wll receve label Hence, e () e () = ne r ne Thus n all the possbltes, the constructed supergraph H satses the condtons or dvsor cordal graph That s, any graph G can be embedded as an nduced subgraph o a dvsor cordal graph Denton : A graph G s sad to have a magc labelng wth constant C there exsts a bjecton : V( E( {,,, p q} such that ( u) ( v) ( uv) C or all uv E( Kotzg and Rosa [] have ntroduced the concept o magc labelng n 97 and nvestgated magc labelng or some standard graphs Denton : A graph G s sad to have a totally magc cordal labelng wth constant C there exsts a mappng : V( E( {,} such that ( u) ( v) ( uv) C (mod ) or all uv E( provded the condton v e v e () () () () holds In, Caht [8] has ntroduced the concept o total magc cordal labelng as a weaker verson o magc labelng Theorem : Any graph G can be embedded as an nduced subgraph o a total magc cordal graph Proo Label the vertces o graph G n such a way that v () v () and as per denton o total magc cordal labelng assgn edge labels such that ( u) ( v) ( uv) C (mod ) V be the set o vertces wth label and be the cardnalty o set V and C Case : I E respectvely Subcase : When n V n V ( ) ( ) ( ) ( ) (say r ) E be the set o edges wth label whle nv and r The new graph H can be obtaned by addng vertces to the graph G and jon them to arbtrary r member o V Assgn label to these new vertces Consequently edges wll receve label r = n( V ) n( E ) n( V ) n( E ) Thus, v () e () v () e () Subcase : When n V n V ( ) ( ) ( ) ( ) (say r ) r The new graph H can be obtaned by addng vertces to the graph G and jon them to arbtrary r member o V Assgn label to these new vertces Consequently edges wll receve label r = n( V ) n( E ) n( V ) n( E ) Thus, v () e () v () e () Case : I C Subcase : When n V n V ( ) ( ) ( ) ( ) (say r ) DOI: 979/ wwwosrjournalsorg 83 Page

5 Ombeddng and NP-Complete Problems o Equtable Labelngs r The new graph H can be obtaned by addng vertces to the graph G and jon them to arbtrary r member o V Assgn label to these new vertces Consequently edges wll receve label r = n( V ) n( E ) n( V ) n( E ) Thus, v () e () v () e () Subcase : When n V n V ( ) ( ) ( ) ( ) (say r ) r The new graph H can be obtaned by addng vertces to the graph G and jon them to arbtrary r member o V Assgn label to these new vertces Consequently edges wll receve label r Thus, v () e () v () e () = n( V ) n( E ) n( V ) n( E ) Thus n all the possbltes, the constructed supergraph H satses the condtons or total magc cordal graph That s, any graph G can be embedded as an nduced subgraph o a total magc cordal graph III NP-Complete Problems Theorem 3: Any planar graph G can be embedded as an nduced subgraph o a planar total edge product cordal (derence cordal, total sequental cordal, dvsor cordal, total magc cordal) graph Proo I G s planar graph Then the supergraph H constructed n Theorem (correspondng theorems 4, 7, 9 and ) s a planar graph Hence the result Theorem 3: Any connected graph G can be embedded as an nduced subgraph o a connected total edge product cordal (derence cordal, total sequental cordal, dvsor cordal, total magc cordal) graph Proo I G s connected graph Then the supergraph H constructed n Theorem (correspondng theorems 4, 7, 9 and ) s a connected graph Hence the result Theorem 33: The problem o decdng whether the chromatc number ( k, where k 3, s NP-complete even or total edge product cordal (derence cordal, total sequental cordal, dvsor cordal, total magc cordal) graph Proo G be a graph wth chromatc number ( 3 supergraph H constructed n Theorem s total edge product cordal (correspondng theorems 4, 7, 9 and ), whch contans G as an nduced subgraph Then obvously we have ( H) ( H) Snce the problem o decdng whether the chromatc number ( k, where k 3, s NP-complete by [] It ollows that decdng whether the chromatc number ( k, where k 3, s NP-complete even or prme cordal graphs Theorem 34: The problem o decdng whether the clque number ( k s NP-complete even when restrcted to total edge product cordal (derence cordal, total sequental cordal, dvsor cordal, total magc cordal) graph Proo Snce the problem o decdng whether the clque number o a graph ( k s NP-complete by [] and ( H) ( or the supergraph H constructed n Theorem s total edge product cordal (correspondng theorems 4, 7, 9 and ) Hence the result Theorem 35: The problem o decdng whether the domnaton number (total domnaton number) s less than or equal to k s NP-complete even when restrcted to total edge product cordal (derence cordal, total sequental cordal, dvsor cordal, total magc cordal) graph Proo Snce the problem o decdng whether the domnaton number (total domnaton number) o a graph G s less than or equal to k s NP-complete as reported n [] and the supergraph H constructed n Theorem s total edge product cordal (correspondng theorems 4, 7, 9 and ) has domnaton number greater than or equal to domnaton number o G Hence the result DOI: 979/ wwwosrjournalsorg 84 Page

6 Ombeddng and NP-Complete Problems o Equtable Labelngs IV Concludng Remarks Acharya et al [] have proved that any graph G can be embedded as an nduced subgraph o a graceul graph Thus showng the mpossblty o obtanng any orbdden subgraph characterzaton or graceul graphs on the same lne Acharya et al [3, 4], Anandavally et al [5], Germna and Ajtha [6], Vadya and Vhol [7] and Vadya and Barasara [8] have dscussed embeddng and NP-Complete problems n the context o varous graph labelng schemes However here we have dscussed embeddng and NP-Complete problems or total edge product cordal labelng, derence cordal labelng, total sequental cordal labelng, dvsor cordal labelng and total magc cordal labelng Acknowledgement The authors are hghly thankul to the anonymous reeree or ther knd suggestons and comments Reerences [] J A Gallan, A Dynamc Survey o Graph Labelng, The electronc journal o combnatorcs, 7, 4, #DS6 [] I Caht, Cordal Graphs: A weaker verson o graceul and harmonous graphs, Ars Combnatora, 3, 987, - 7 [3] S K Vadya and C M Barasara, Edge Product Cordal Labelng o Graphs, Journal o Mathematcal and Computatonal Scence, (5),, [4] G Chartrand and L Lesnak, Graphs and Dgraphs 4/e (Chapman & Hall/CRC Press, 5) [5] S K Vadya and C M Barasara, Total edge product cordal labelng o graphs, Malaya Journal o Matematk, 3(), 3, [6] R Ponraj, S Sathsh Narayanan and R Kala, Derence Cordal Labelng o Graphs, Global Journal o Mathematcal Scences: Theory and Practcal, 5(3), 3, [7] D W Bange, A E Barkauskas and P J Slater, Smply sequental and graceul graphs, Proceedng o Tenth Southeastern Conerence on Combnatorcs, Graph Theory and Computng, Utltas Mathematca Publshng Inc, Wnnpeg, Canada, 979 [8] I Caht, Some Totally Modular Cordal Graphs, Dscussones Mathematcae Graph Theory,,, [9] R Varatharajan, S Navanaeethakrshnan and K Nagarajan, Dvsor cordal graphs, Internatonal Journal o Mathematcal Combnatorcs, 4,, 5-5 [] A Kotzg and A Rosa, Magc valuatons o nte graphs, Canadan Mathematcal Bulletn, 3(4), 97, [] M R Garey and D S Johnson Computers and Intractablty- A gude to the theory o NP-Completeness (W H Freeman and Company, 979) [] B D Acharya, S B Rao and S Arumugam, Embeddngs and NP-Complete problems or Graceul Graphs, Labelng o Dscrete Structures and Applcatons (B D Acharya, S Arumugam and A Rosa (eds)), Narosa Publshng House, New Delh, 8, 57-6 [3] B D Acharya, K A Germna, K L Prncy and S B Rao, Graph Labelngs, Embeddngs and NP-Completeness theorems, Journal o Combnatoral Mathematcs and Combnatoral Computaton, 67, 8, 63-8 [4] B D Acharya, K A Germna, K L Prncy and S B Rao, Embeddngs and NP-Completeness theorems or set-graceul, topologcally set-graceul and set-sequental graphs, Dscrete Mathematcal Scences and Cryptography, (4), 9, [5] T M K Anandavally, S Arumugam, K A Germna and S B Rao, On Sum Composte Graphs and Embeddng Problems, Journal o Combnatoral Mathematcs and Combnatoral Computng, 69, 9, 3-3 [6] K A Germna and V Ajtha, Embeddng Theorems and NP-Completeness, Proceedngs o the Natonal Conerence on Operator Theory and Applcatons, 9, 97-3 [7] S K Vadya and P L Vhol, Embeddng and NP-Complete Problems or 3-Equtable Graphs, Internatonal Journal o Contemporary Advanced Mathematcs, (),, - 7 [8] S K Vadya and C M Barasara, Embeddng and NP-complete Problems or Some Equtable Labelngs, Journal o Appled Computer Scence and Mathematcs, 8 (8), 4, 38 4 DOI: 979/ wwwosrjournalsorg 85 Page