On the 2-Domination Number of Complete Grid Graphs

Transcription

1 Ope Joural of Dicrete Mathematic, 0,, -0 ISSN Olie: - ISSN Prit: - O the -Domiatio Number of Complete Grid Graph Ramy Shahee, Suhail Mahfud, Khame Almaea Departmet of Mathematic, Faculty of Sciece, Tihree Uiverity, Lattakia, Syria How to cite thi paper: Shahee, R, Mahfud, S ad Almaea, K (0) O the -Domiatio Number of Complete Grid Graph Ope Joural of Dicrete Mathematic,, -0 Received: November, 0 Accepted: Jauary 0, 0 Publihed: Jauary, 0 Abtract A et D of vertice of a graph G = (V, E) i called k-domiatig if every vertex v V D i adacet to ome k vertice of D The k-domiatio umber of a graph G, k ( G), i the order of a mallet k-domiatig et of G I thi paper we calculate the k-domiatio umber (for k = ) of the product of two path P m P for m =,,,, ad arbitrary Thee reult were how a error i the paper [] Copyright 0 by author ad Scietific Reearch Publihig Ic Thi work i liceed uder the Creative Commo Attributio Iteratioal Licee (CC BY 0) Ope Acce Keyword k-domiatig Set, k-domiatio Number, -Domiatig Set, -Domiatio Number, Carteia Product Graph, Path Itroductio = be a graph A ubet of vertice D V i called a -domiatig et of G if for every v V, either v D or v i adacet to at leat two vertice of D The -domiatio umber ( G) i equal to mi { D : D i a domiatig et of G The Carteia product G H of two graph G ad H i the graph with vertex et V ( G H) = V ( G) V ( H), where two vertice ( v, v ), ( u, u) G H are adacet if ad oly if either vu E( G) ad v = u or vu E( H) ad v = u Let G be a path of order with vertex et V ( G) = {,,, The for two path of order m ad repectively, we have Pm P = {( i, ) : i m, The th colum of P m P i K = {( i, ) : i =,, m If D i a -domiatig et for P m P, the we put W = D K Let = W The equece (,,, ) i called a -domiatig equece correpodig to D For a graph G, we refer to miimum ad maximum degree by δ ( G) ad ( G), ad for implicity deoted thoe by δ ad Δ, repectively Alo, we deote by V ad E to order ad ize of graph G, repectively Let G ( V, E) DOI: 0/odm000 Jauary, 0

2 R Shahee et al Notatio ad Termiology Fik ad Jacobo [] [] i 98 to itroduced the cocept of multiple domiatio A ubet D V i k-domiatig i G if every vertex of V D ha at leat k eighbor i D The cardiality of a miimum k-domiatig et i called the k-domiatio umber k ( G) of G Clearly, g ( G) = g( G) Naturally, every k-domiatig et of a graph G cotai all vertice of degree le tha k Of coure, every ( k + ) -domiatig et i alo a k-domiatig et ad o k ( G) k + ( G) Moreover, the vertex et V i the oly ( + ) -domiatig et but evidetly it i ot a miimum -domiatig et Thu every graph G atifie k ( ) ( ) ( ) ( ) G G G < G = V k+ + For a compreheive treatmet of domiatio i graph, ee the moograph by Haye et al [] Alo, for more iformatio ee [] [] Fik ad Jacobo [], itroduced the followig theorem: Theorem [] If k, i a iteger ad G i a graph with k ( G), the k ( G) ( G) + k T + Theorem [] If T i a tree, the ( T ) I [], Haberg ad Volkma, proved the followig theorem Theorem [] Let G = ( V, E) be a graph of order ad miimum degree δ ad i δ + V k δ let k N If k, the k ( G) kl ( δ + ) + k l ( δ + ) δ + i = 0 i! ( δ + ) Cockaye, et al [], etablihed a upper boud for the k-domiatio umber of a graph G ha miimum degree k, they gave the followig reult Theorem [] Let G be a graph with miimum degree at leat k, the kv k ( G) k + ( ) Blidia, et al [8], tudied the k-domiatio umber They itroduced the followig reult Theorem [8] Let G be a bipartite graph ad S i the et of all vertice of degree at V + S mot k, the k ( G) Favaro, et al [9], gave ew upper boud of k ( G) Corollary [9] Let G be a graph of order ad miimum degree δ If k δ i δ a iteger, the k ( G) V δ + k I [], Haye et al howed that the -domiatio umber i bouded from below by the total domiatio umber for every otrivial tree Theorem [] For every otrivial tree, ( T) t ( T) Alo, Volkma [0] gave the importat followig reult Theorem 8 [0] Let G be a graph with miimum degree δ k +, the V + k ( G) k + ( G) Shahee [] coidered the -domiatio umber of Toroidal grid graph ad gave

3 R Shahee et al a upper ad lower boud Alo, i [], he itroduced the followig reult Theorem 9 [] ) ( C ) = ) ( C C ) = : 0( mod ), ( C C ) = + :,( mod) ) ( C C ) = + : 0,,( mod8), ( C C ) = + + :,,,, ( mod) ) ( C C ) = ) ( C C ) = : 0( mod ), ( C C ) = + :,( mod) ) ( C C ) = : 0,,( mod), ( C C ) = + :,,, 8, 9,0( mod) ( C C ) = + ( ), :,,,, mod I thi paper we calculate the k-domiatio umber (for k = ) of the product of two path Pm P for m =,,,, ad arbitrary Thee reult were how a error i the paper [] We believe that thee reult were wrog I our paper we will provide improved ad corrected her, epecially for m =,, The followig formula appeared i [], ( ) ( ) ( ) ( ) ( ) P = + P P = P P = P P = ( ) ( ) ( ) P P = Pk+ P = k + ( Pm P) = m m ( ) ( P P ) = m : m 0( mod ) m : mod, I thi paper, we correct the reult i [] ad prove the followig: Mai Reult ( ) ( ) ( ) ( ) P = + P P = P P = + ( P P ) = :,( mod8 ), ( P P ) = + : 0,,,,, ( mod 8 ) ( P P ) = + :,,,( mod ), ( P P ) = + + : 0,, ( mod ) Our mai reult here are to etablih the domiatio umber of Carteia product of two path P m ad P for m =,,,, ad arbitrary We tudy -domiatig et i complete grid graph uig oe techique: by give a miimum of upper -domiatig et D of Pm P ad the we etablih that D i a miimum -domiatig et of P m P for everal value of m ad arbitrary Defiitely we have ( Pm P) = D Let G be a path of order with vertex et V ( G) = {,,, For two path of order m ad repectively i: Pm P = {( i, ) : i m, The th colum Pm P i K = {( i, ) : i =,, m If D i a -domiatig et for Pm P the we put W = D K Let = W The

4 R Shahee et al equece ( ),,, i called a -domiatig equece correpodig to D Alway we have, m Suppoe that = 0 for ome (where or ) The vertice of the th colum ca oly be -domiated by vertice of the ( )t colum ad ( + )t colum Thu we have + + = m, the = + = m I geeral m Notice ) The tudy of -domiatig equece (,,, ) i the ame a the tudy of the -domiatig equece (,,, ) ) If ubequece (, +,, + k) i ot poible, the it revere ( + k,, +, ) i ot poible ) We ay that two ubequece (,, + q),( + q+,, + r) are equivalet, if the equece (,, + q, + q+,, + r) i poible We eed the ueful followig lemma Lemma There i a miimum -domiatig et for Pm P with -domiatig equece (,,, ) uch that, for all =,,,, i m m Proof Let D be a miimum -domiatig et for P m P with -domiatig equece (,,, ) Aume that for ome, i large The we modify D by movig two vertice from colum, oe to colum ad aother oe to colum +, uch that the reultig et i till -domiatig et for P m P For i m ad, let W = D {( i, ),( i+, ),( i+, ),( i+, ) If W =, the we defie D = ( D W) {( i, ),( i+,, ) ( i+, +, ) ( i+, ), ee Figure We repeat thi proce if eceary evetually lead to a -domiatig et with required propertie Alo, we get D i a -domiatig et for Pm P with D = D Thu, we ca aume that every four coecutive vertice of the th colum iclude at mot three vertice of D Thi implie that m, for all To prove the lower boud, we uppoe that K D i be a maximum, ie, = m The for each ( i, ) D, we have {( i, + ),( i, + ),( i+, + ) D Whe = m, there at mut m m = m vertice doe ot i K D Thi implie that + m So, the ame a for m By Lemma, alway we have a miimum -domiatig et D with -domiatig equece (,,, ), uch that m m, for all =,,, + Lemma ( P ) = Figure Modify D

5 R Shahee et al D D = k ; k Proof Let ( ) We have D i a -domiatig et of P for ( mod ) with {( ) i a -domiatig et of P for 0( mod ) with ( ) + D =, alo + D { = Let D be a miimum -domiatig et for P with V ( P) { x, x,, x E( P) x x D x D x, x = Sice xx, we eed to,, alo if the + are belog to D, thi implie that x D for Thu implie that + D + = We reult that + ( P ) = Theorem ( ) P P = D,k : k =, k : k Proof Let a et ( ) ( ) It i clear that D = () We ca check that D i -domiatig et for P P, ee Figure Let D be a miimum -domiatig et for P P with domiatig equece (,, ) If i for all =,,, the D = () Let = 0 for ome, the = + =, alo we have ad Now we defie a ew equece (,, ), (ot ecearily a -domiatig equece) a follow: For =, if = or, we put =, = + ad = + If or, we put =, = + ad + = + + Otherwie = We get a equece (,, ) have property that each with By (), () ad () i ( ) = () = = D = = P P = Thi complete the proof of the theorem P P = + D = (, k ) : k (, k) : k (,k ),(,k ) : k Theorem ( ) Proof Let Figure A -domiatig et for P P0

6 R Shahee et al We have D = (,k ),(,k ) : k (,k ) : k (, k) : k D = + ad D = + () By defiitio D ad D we ote that D i -domiatig et for P P whe = 0,( mod), (ee Figure, for P P ) D i -domiatig et for P P whe = ( mod ), (ee Figure, for P P 0 ) Let D be a miimum -domiatig et for P P with -domiatig equece (,, ) we have, ad if, = the,, if, = the, Alo for < <, if = 0 the = + =, = the + +, = the + +, If o oe of = 0 for all, the D = + = () Let = 0 (,,, (ot ecearily a -domiatig equece ) a follow: If =, the we put =, = + ad + = + +, otherwie \ = We have D = =,, have the or ) for ome, we defie a equece ( ) We ote that the equece ( ) = = property if = the + + Thu implie that D = + = () From (), () ad () we get the required reult :,( mod8 ), Theorem ( P P ) = + : 0,,,,, ( mod 8 ) Figure A -domiatig et for P P Figure A -domiatig et for P P0

7 R Shahee et al Proof Let a et D defied a follow: D = { (, ),(,) (, k ),(, k ); k (,8k ); k 8 (,8k ),(,8k ),(,8k ); k 8 (,8k ); k (,8k ); k 8 8 (,8 k),(,8 k),(,8 k) ; k (,8k + ); k 8 8 {(, ), {(, ) D = D = We ca check that the followig et are -domiatig et for P P (ee Figure, P P ) a idicated: for D i -domiatig et for P P D D i -domiatig et for P P D D i -domiatig et for P P We have whe 0, ( mod 8) whe,,( mod8) whe,,( mod8) :,( mod8 ), D = :,,,( mod8 ), + : 0, ( mod 8 ) Let D be a miimum -domiatig et for P P with -domiatig equece,, we hall how that ( ) D :,( mod8 ), = + : 0,,,,, ( mod 8 ) By Lemma, we have Thu If = the + + If = the + + If = the + + Alo, we have, If, = the,, ad if, = the Figure A -domiatig et for P P 8

8 R Shahee et al, We defie a ew et equece) a follow: if, let, the we put D with equece ( ) M,,, (ot ecearily a -domiatig = Now, for = to =, if = M, M M = + ad + = + + Thu, for, we have Sice if the ad if =, the + + = thi implie that M + M + = =, which implie M M + that = + + = + = We have three cae: Cae :,, the,, thee implie that + ad + alo 8 8 ( ) D = = = = = = = Cae :, = the, Thu implie that, = ad, + The 8 D = = = = = = = Cae : =, = ad, or =, = ad, Two cae are imilar by ymmetry We coider the firt cae: =, ad =,, thi implie that = +, =, =, = + ad 8 8 D = = = ( ) = = = = But, we have the -domiatio umber i poitive iteger umber, alo we have = + for,( mod8), + For 0, ( mod8 ), For, ( mod8 ), + = + + For, ( mod8 ), Thu implie that D ;,( mod8 ), + ; 0,,,,, ( mod 8 ), 9

9 R Shahee et al Fially, we get ( P P ) = ( ) ( P P ) = + ( ) Thi complete the proof of the theorem Theorem ( P P ) Proof Let a et D defied a follow: { :, mod8, : 0,,,,, mod 8, + :,,,( mod ), = + + : 0,, ( mod ) {(, ),(,) {(, ),(, ),(, ) : ( mod ) { (, ) : ( mod ) { (, ),(, ),(, ) : ( mod ) { (, ),(, ) : ( mod ) { (, ),(, ) : ( mod ) { (, ),(, ) : 0( mod ) { (, ),(, ) : ( mod ) ad D = We ca check that the followig et are -domiatig et for P P (ee Figure, P P ) a idicated: for We have D + ad { D { K D { ( ) ( ) ( ) ( ) D { (, ) : 0, ( mod ) D: ( mod ) { D { K D { (, ),(, ) :, ( mod ) { D { K D { ( ) ( ) ( ) ( ) ( P P ),,,,, : mod,,,,, : mod + :,,,( mod ), + + : 0,, ( mod ) Thi complete the proof of the theorem Lemma The followig cae are ot poible: ) (,,, ) ) (,, ) ) (,,, ) Figure A -domiatig et for P P 0

10 R Shahee et al ) (,,,,, ) ) (,, ) ) (,,,,, ) Proof It follow directly from the drawig Lemma ) There i oe cae for ubequece (, +, +, +, + ) = (,,,, ) ) There i oe cae for ubequece (, +, +, + ) = (,,, ) ) There i oe cae for ubequece (, +, +, +, + ) = (,,,,) ) There i oe cae for ubequece (, +, +, +, + ) = (,,,,) Proof It follow directly from the drawig (ee Figure ) Lemma ) ) ) ) If = the ) If = the + + Proof ) By Lemma, imply that ) By, we have + 8 If = 8, the we have the cae (,,, ) (,,, ),(,,, ),(,,, ),(,,, ),(,,,) = From Lemma, we have +, thi implie that If the Thi implie that ) We have + ad (,, ) (,,) + + If = (where the cae ( ) ( ) cae (,, ) i ot compatible with ay of the cae whe + =, the there i oe cae,,,,, are ot poible) But the + = 8, thi implie that Figure Cae,, ad of Lemma

11 R Shahee et al The the + (where the cae (,,,,, ) i ot poible) If = = + ) We have, the from i ) We have, the from i Lemma If =, the = or + = Thi complete the proof of the Lemma Proof We uppoe the cotrary, + From Lemma,, + <, ele + Now, we mut tudy the cae = + = We have + + = 0, by Lem- ma, the cae (,,,,, ) i ot poible, thi implie that ot all elemet of the ubequece ( +,, + ) are equal to the value If,,,, where at leat oe of them i equal or greater tha, the, thi i a cotradictio with + met ( +,, + ) = Now, we have = 0, where oe of the ubequece ele- i at mot equal the value (where ) We coider the = for + + : + = or + = (where two cae are imilar), we tudy the cae + = + =, thee implie that + + = By Lemma, we have cae ) the + 8 the , thi i a cotradictio ) + = or + = (where two cae are imilar), we tudy the cae + = the +, (becaue the cae (,, ) i ot poible) If + = the + ad we have + = the poible) Thu implie that + + (becaue two cae (,, ),(,, ) are ot =, thi i a cotradictio ) + =, the we have two ubcae reult from : Subcae : + = + = the +, + (becaue two cae, +, + =,, ad ( +, +, + ) = (,, ) are ot poible) Thu implie ( ) ( ) that +, thi i a cotradictio Subcae : If + =, + = or coverely (two cae are imilar i tudyig), o we will tudy cae + =, + = the +, if +, the, becaue + + 8, we have The If + =, the = We have + + +, thi i a cotradictio) 8 Thi implie that

12 R Shahee et al + thi i a cotradictio Fially, we get if + + =, the = or + = Thi completely the proof Reult If (, +, +, +, +, +, + ) : a: (,,,,,, ), a :(,,,,,, ), a : (,,,,,, ), a : (,,,,,, ), a : (,,,,,, ), a :(,,,,,, ), a : (,,,,,, ), a8 : (,,,,,, ), a9 : (,,,,,, ), a 0 :(,,,,,, ), a :(,,,,,, ), a :(,,,,,, ), a : (,,,,,, ), a : (,,,,,, ), a : (,,,,,, ) =, the from Lemma, we have the cae for ubequece It i cae (where = with + = ) We have three cae with + =, + = ad + = Cae : + = (icludig the cae = ad + = or + = ad + = ) We have thee cae are a, a, a, a, a8, a, a ad come before thee cae, = or come after thee cae + =, ie, if =, + = the = ad if + =, + = the + = Cae : + =, + = ad thee are the 8 remaiig cae We will tudy thee cae after reectig iomorphim cae whe there i two cae or more, where (,, + ) = ( +,, ), the we will tudy oly oe cae We have 8 cae a follow: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a :,,,,,,, a :,,,,,,, a :,,,,,,, a9 :,,,,,,, a :,,,,,,, a :,,,,,,, a :,,,,,,, a :,,,,,, 0 We ote that two cae a, a are imilar where oe of them i cotrary to the other oe, o we tudy the cae a Alo, two cae a, a are imilar, o we tudy the cae a The we tudy thee cae: a, a, a, a9, a0, a Notice We ote that all the poible cae i Reult, do ot begi or ed with or ad it do ot begi or ed with + + or uch that = or + =, ad + = or + = Thu implie that if =, + =, + the Alo, we ote cae a, a, a are begiig with (,,, ), but from Lemma, we get = Now, remai our three cae for tudyig by the followig lemma are: ( ) ( ) ( ) a :,,,,,,, a :,,,,,,, a :,,,,,, 9 0 = = + where = {( ) or {( ) =, alo for ( ) cae = {( ) or {( ) + + Lemma If =, uch that + =, + =, the Reult If, k, k =, the k = {, or k = {(, ) becaue it are imilar to two k, k =,, repectively + Furthermore,

13 R Shahee et al if + = the Proof By Reult, if k = {(, + + ), k {( ), = {(, + + ) or k {( ) = +, + the + Aume k {(, ) k get + = +, + + = From Lemma, we = + the we have two cae for k + + : Cae k {(, ),(, ),(, ) + + = The + + =, by lemma, Cae k {(, ),(, ),(, ) + {(, ),(, ),(, ) = or + k = ad both cae are imilar, o we will coider the firt cae We have + the by Lemma, + Aume + =, if If + = the + the proof i fiih Aume the we have cae + 8 =,, or Subcae If + 8 = the + 9 Thi implie that = = that {By Lemma, + 8 Subcae If + 8 = the If + > 9 the = 9 the we have oly oe cae (,, 9 ) (,,,, + + = ) or ( + 9,, + ) = (,,,,) For ay cae we have + 8 = So, we get Which implie that + + Subcae If + 8 = the + 9 = {becaue the cae +, +, +, + 8, + 9 =,,,, i ot poible, by Lemma The ( ) ( ) = = Aume Subcae If + 8 = the + =, + 8 =, we have the followig cae: + 9 the 9 that = {becaue there i oly oe cae for (,, ) (,,) 8 9 > = uch { K + K + 8 K + 9 S= {(, +,, ) ( +,, ) ( +,, ) ( + 8,, ) ( + 8,, ) ( + 9) But accordig to ditributio vertice k + S ad k+ S we have k + {(, ),(, ),(, ) = the = Thi implie that + + +

14 R Shahee et al (, 8, 9) (,,) = We will tudy the cae that lead to = 8, {becaue the cae which lead to + + we have the fixed cae (,, ) (,,) + + =, ie, the proof will be doe Now, = We will coider the vertice k+ 0 S which imply the followig: If 0 + = the ( ),,,, +, +, +, thi implie that ad ( +, +, + ) = (,,) i ot poible If + 0 = the (,,,,,, ) ad that (,, ) (,, ),(,, ),(,, ) i (,, ) Thu implie that (,, ) (,,,,,,) = = which imply = or (,, ), ad the oly poible cae By Lemma ad Lemma i + =, thee implie that If 0 for +, +, + : + + = = the (,,,,,, ) + {becaue the cae ( ) ) If + = the + = ad ) If + = ad + = the ) If + =, + = ad + get + = ad , ie, + + = We have,, i ot poible The we have the followig cae + =, but the cae ( ) + =, alo the cae ( ),, i ot poible,, i ot poible,, + =,,,,,, which + = the ( ) ( ) ) If + = ad + = the,,,,,, i ot poible If + =, + = ad + = the we get,, + =,,,,,, Durig the proof of Lemma, we otice that if = ( ) ( ) + 8 ad + =, the + + =, but the cae ( ) Thi complete the proof Reult Baed o the Lemma, ad the other Lemma ad reult precede it We ee that whe we have cae of =, the the oly cae that come after it, i = uch that (,, ) (,,,,,, + + = ) which cotiue i the ame way or it i followed by colum cotai vertice from S {by Lemma, + 0, becaue + =, + = Whe thi cae i repeated the + ad the whe the cae + = it i eceary, the cae + + q + r + + q well {where + + q + r thee implie that = the ( P P) = + = Lemma 8 Let S be -domiatig et for Ρ Ρ the: = exit a

15 R Shahee et al ) ad (, ) + + ) If + = the + + = 8 ( + = the + + = 8 ) ) ( ) ) ) the ) ) 9 9 = = 0 0 ad if = = = = = = = 8) If + = the either, alo if + = the either or = = = = 0 the = or = =, alo if = = 0 Proof The tudy of domiatig equece (,,, ) the domiatig equece (,,, ), o we tudy oe cae ( ) tudy of r i the ame a the tudy of = = r+ i the ame a the tudy of,,, ) We have, if = the thu, + if the ( ) thee implie that + ) If + =, the we have oly oe the cae k {(, ),(, ),(,) implie that k = {(, ) ad = the + + = 8 i Alo, the = thee ) If +, the {becaue ad if + = the by, = 8 = ) If + = the the cae: = = = 8 thee implie that 9 ad if + = 9 {becaue + Aume that + =, the we have three +, becaue the cae ( ) ( ) poible Alo the cae ( ) ( ) k = {(, ),(, ),(,) ad thi i ot poible ) =, = the + becaue the cae (,, ) = (,,), (,, ) = (,, ) are ot poible =, = the +, becaue the cae (,,, ) (,,,) ) =, = the,, =,, i ot,, =,, i ot poible, ele whe ) (,,, ) (,,, ) = =, = are ot poible Thu implie that we have 9 =

16 R Shahee et al ) By Lemma, we have two cae for ( ) ( ) = 9 ad thee two cae are,,,,,,,,,, furthermore thee caot be how here becaue Thu implie that we 0 = ) If + the (where by Lemma, we have = 8 thee implie that = = = = + = = = 8 + Thu implie that 8 ) Let + = the = = 8 + = {becaue + ) If the from Lemma, Let = {becaue > the Thi implie that = 8) If + = the either = {by Notice or = We have + =, the we have three cae: 8) =, =, the + + becaue the cae,,,, =,,,,,,,,,,,,,, or,,,, are ot poible ( ) ( ) ( ) ( ) ( ) Thu implie that = 8) =, =, the (,,,, ) (,,,,) = 0 ad if = = 0 the = By Lemma, = Thu implie that = =, the ( ) cae ( ) ( ) ( ) ( ) ( ) = ( ) i ot compatible with the cae ( ) ( ) Thi complete the proof = = = 8),,,,, it ha miimal umeral i the followig,,,, =,,,,,,,,, or,,,, ad for the cae,,,,, =, Thu implie that Theorem ( P P ) + :,,,( mod ), = + + : 0,,( mod ) Proof By Reult, we have ( p p) = = By Theorem, we get ( p p ) = + :,,,( mod ) Now, for 0,, ( mod ), by Theorem, we have ( p p ) + +

17 R Shahee et al From Reult, we have ( ) p p + We will tudy the cae: ) 0( mod ) We have ( ) p p = So, we coider the followig: = a) + = the + + = 8 ad by Lemma 8, ( p p) ( ) = = = ( ) = = , + p p + + = + = b) + if + the = = ( p p) ( ) = = = + = + = + + Let + = the by Lemma 8, or = = If the = = = ( p p) ( ) = = = = + + = + + {where the cae + = i the ame a + = If 8, we have = < the by Lemma = = = ( p p) ( ) = = = + = + = + + : 0 mod Ad with Theorem, we get ( p p ) = + + ( ) ) Whe ( mod ) we have two cae: a) + = Thu implie that + + = 8 the ( ) = = = ( ) p p = = = + = + + b) + {where + the = = ( p p) ( ) = = + 0 = + = + + 8

18 R Shahee et al : mod The by Theorem, we get ( p p ) = + + ( ) ) ( mod ) We have two cae: a) If + = the + + = 8 Thu implie that = = = + + ( p p) ( ) = = b) If + the + Thu implie that = = = + = + + ( p p) ( ) = = = : mod Fially, we get By Theorem, we get ( p p ) = + + ( ) ( p p ) Thi complete the proof Referece + :,,,( mod ), = + + : 0,, ( mod ) [] Moha, JJ ad Kelkar, I (0) Retraied -Domiatio Number of Complete Grid Graph Iteratioal Joural of Applied Mathematic ad Computatio,, -8 [] Fik, JF ad Jacobo, MS (98) -Domiatio i graph, i: Graph Theory with Applicatio to Algorithm ad Computer Sciece Joh Wiley ad So, New York, 8-00 [] Fik, JF ad Jacobo, MS (98) O -Domiatio, -Depedece ad Forbidde Subgraph I: Graph Theory with Applicatio to Algorithm ad Computer Sciece, Joh Wiley ad So, New York, 0- [] Haye, TW, Hedetiemi, ST, Heig, MA ad Slater, PJ (00) H-Formig Set i Graph Dicrete Mathematic,, [] Haye, TW, Hedetiemi, ST ad Slater, PJ (998) Fudametal of Domiatio i Graph Marcel Dekker, Ic, New York [] Haberg, A ad Volkma, L (009) Upper Boud o the k-domiatio Number ad the k-roma Domiatio Number Dicrete Applied Mathematic,, -9 [] Cockaye, EJ, Gamble, B ad Shepherd, B (98) A Upper Boud for the k-domiatio Number of a Graph Joural of Graph Theory, 9, - [8] Blidia, M, Chellali, M ad Volkma, L (00) Some Boud o the p-domiatio Number i Tree Dicrete Mathematic, 0, [9] Favaro, O, Haberg, A ad Volkma, L (008) O k-domiatio ad Miimum Degree i Graph Joural of Graph Theory,,

19 R Shahee et al [0] Volkma, L (00) A Boud o the k-domiatio Number of a Graph Czecholovak Mathematical Joural, 0, -8 [] Shahee, R (009) Boud for the -Domiatio Number of Toroidal Grid Graph Iteratioal Joural of Computer Mathematic, 8, [] Shahee, R (0) O the -Domiatio Number of Carteia Product of Two Cycle Advace ad Applicatio i Dicrete Mathematic,, 8-08 Submit or recommed ext maucript to SCIRP ad we will provide bet ervice for you: Acceptig pre-ubmiio iquirie through , Facebook, LikedI, Twitter, etc A wide electio of oural (icluive of 9 ubect, more tha 00 oural) Providig -hour high-quality ervice Uer-friedly olie ubmiio ytem Fair ad wift peer-review ytem Efficiet typeettig ad proofreadig procedure Diplay of the reult of dowload ad viit, a well a the umber of cited article Maximum diemiatio of your reearch work Submit your maucript at: Or cotact odm@cirporg 0