Dominating Sets and Domination Polynomials of Square Of Cycles

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1 IOSR Joural of Mathematics IOSR-JM) ISSN: Volume 3, Issue 4 Sep-Oct. 01), PP Domiatig Sets ad Domiatio Polyomials of Square Of Cycles A. Vijaya 1, K. Lal Gipso 1 Assistat Professor, Departmet of Mathematics, Nesamoy Memorial Christia College, Marthadam, Kaayakumari District, Tamil Nadu, South Idia. Assistat Professor, Departmet of Mathematics, Mar Ephraem College of Egieerig ad Techology, Mathadam, Kaayakumari District, Tamil Nadu, South Idia. Abstract: Let G = V, E) be a simple graph. A set S V is a domiatig set of G, if every vertex i V-S is adjacet to atleast oe vertex i S. Let C be the square of the Cycle C ad let D C, i ) deote the family of all domiatig sets of C with cardiality i. Let d C, i ) = D C, i ). I this paper, we obtai a recursive formula for d C, i ). Usig this recursive formula, we costruct the polyomial, D C x =, ) i d C, ) i x i, which we call domiatio polyomial of this polyomial. Keywords: domiatio set, domiatio umber, domiatio polyomials. C ad obtai some properties of I. Itroductio Let G = V,E) be a simple graph of order V =. For ay vertex vv, the ope eighbourhood of is the set N v) = {uv/uve} ad the closed eighbourhood of is the set N[v] = N v) {v}. For a set S V, the ope eighbourhood of S is NS) = U S N v) ad the closed eighbourhood of S is N[S] = NS) S. A set S V is a domiatig set of G, if N[S] = V, or equivaletly, every vertex i V-S is adjacet to atleast oe vertex i S. The domiatio umber of a graph G is defied as the miimum size of a domiatig set of vertices i G ad it is deoted as G.A cycle ca be defied as a closed path, ad is deoted byc Defiitio The square of a graph: The d power of a graph with the same set of vertices as G ad a edge betwee two vertices if ad oly if there is a path of legth atmost betwee them. Let C be the square of the cycle C d power) with vertices. Let D C, i ) be the family of domiatig sets of the graph C with cardiality i ad let d C, i ) = D C, i ). We call the polyomial i D C, x ) = d C, i) x the domiatio polyomial of the graph C. i I the ext sectio, we costruct the families of the domiatig sets of the square of cycles by recursive method. As usual we use x for the largest iteger less tha or equal to x ad x for the smallest iteger greater tha or equal to x. Also, we deote the set {1, } by [], throughout this paper. II. Domiatig Sets Of Square Of Cycles Let D C, i ) be the family of domiatig sets of C with cardiality i. We ivestigate the domiatig sets of C. We eed the followig lemma to prove our mai results i this sectio. Lemma..1 C ) = By Lemma.1 ad the defiitio of domiatio umber, oe has the followig lemma: 4 Page

2 Domiatig Sets Ad Domiatio Polyomials Of Square Of Cycles Lemma.. D C, i ) = if ad oly if i or i A simple path is a path i which all its iteral vertices have degree two, ad the ed vertices have degree oe. The followig lemma follows from observatio. Lemma.3 [] If a graph G cotais a simple path of legth k -1, the every domiatig set of G must cotai at least k vertices of the path. I order to fid a domiatig set of C, with cardiality i, we oly eed to cosider D C, i 1), D C, i 1), D C, i 1), D C, i 1) ad D C, i 1).The families of these domiatig sets ca be empty or otherwise. Thus, we have eight combiatios, whether these five families are empty or ot. Two of these combiatios are ot possible Lemma. i), & ii)). Also, the combiatio that D C, i 1) D C, i 1) D C, i 1) D C, i 1) D C, i 1) 1 3 4, ) does ot eed to be cosidered because it implies that to cosider five combiatios or cases. We cosider those cases i theorem.7). D P i = See lemma. iii)). Thus we oly eed Lemma.4 If Y D C6, i 1), ad there exists x [] such that Y {x} D C, i ), the Y D C, i 1). : Suppose that Y D C, i 1). Sice Y D C6, i 1), Y cotais at least oe vertex labeled 6, 7 or 8. If 6 Y, the Y D C, i 1), a cotradictio. Hece 7 or 8 Y, but the i this case, Y {x} D C, i ), for ay x [], also a cotradictio. Lemma. i) If D C 1, i 1) = D C3, i 1) = the D C, i 1) = ii) If D C 1, i 1), D C3, i 1) the D C, i 1), iii) If D C 1, i 1) =, D C, i 1) =, D C3, i 1) =, D C4, i 1) =, D C, i 1) =, the D C, i ) = 3 i) Sice D C 1, i 1) = D C3, i 1) =, by lemma., i -1-1 or) i -1. ii) I either case, we have D C, i 1) =. iii) Suppose that D C, i 1) =, so by Lemma., we have i 1 or i 1. If i -1, the i 1 3. Therefore D C3, i 1) =, a cotradictio. iv) Suppose that D C, i ), Let Y D C, i ) ). Thus, atleast oe vertex labeled or 1 or is i Y. If Y, the by Lemma.3, at least oe vertex labeled 1,, 3, 4 or is i Y. If 1 Y or Y, the Y {} D C 1, i 1), a cotradictio. If 3 Y, Page

3 Domiatig Sets Ad Domiatio Polyomials Of Square Of Cycles the Y {} D C, i 1), a cotradictio. Now, suppose that 1 Y. The, by Lemma.3, at least oe vertex labeled, 3, 4, or 6 is i Y. If Y or 3 Y, the Y { 1} D C, i 1) a cotradictio. If 4 Y, the Y { 1} D C3, i 1), a cotradictio. If Y, the Y { 1} D C4, i 1), a cotradictio. If 6 Y, the Y { 1} D C, i 1), a cotradictio. Therefore, D C, i ) =. Lemma. 6 If D C, i ),the i) D C 1, i 1) = D C, i 1) = D C3, i 1) = D C4, i 1) = ad D C, i 1) if ad oly if =k ad i=k for some k N; ii) D C, i 1) = D C3, i 1) = D C4, i 1) = D C, i 1) = the D C 1, i 1) if ad oly if i = ; iii) D C 1, i 1) =, D C, i 1), D C3, i 1), D C4, i 1), k D C, i 1), if ad oly if = k+ ad i = for some kn; iv) D C 1, i 1), D C, i 1), D C3, i 1), D C4, i 1), ad D C, i 1) = if ad oly if i = 3. v) D C 1, i 1), D C, i 1), D C3, i 1), D C4, i 1), ad 1 D C, i 1) if ad oly if +1 i 4. i) ) Sice D C 1, i 1) ) = D C, i 1) = D C3, i 1) = D C4, i 1) =, by lemma., i 1 1 or i 1 If i 1 1, the i ad by lemma., D C, i ) =, a cotradictio. So i + 1, ad sice D C, i ), we have i + 1, which implies that = k ad i = k for some kn. ) If = k ad i = k for some kn, the by lemma., D C 1, i 1) = D C, i 1) = D C3, i 1) = D C4, i 1) =, ad D C, i 1). ii) ) Sice D C, i 1) = D C3, i 1) = D C4, i 1) = D C, i 1) =, by lemma., 3 i 1 or i 1 <. If i 1 3, the i 1 1 ad hece D C, i 1) =, a cotradictio. So i -1. Also, D C 1, i 1). Therefore i -1.Therefore i.but i. Hece i =. 1 iii) ) Sice D C 1, i 1) =, by lemma., i 1 1 or i 1 If i 1 1, the i 1,by Lemma., D C, i 1) = D C3, i 1) = D C4, i 1) = D C, i 1) =, a cotradictio. 6 Page

4 Domiatig Sets Ad Domiatio Polyomials Of Square Of Cycles 1 Therefore i +1. But i 1,because D C, i 1). Therefore i +1. Hece, +1 i k This holds oly if = k + ad i = k+1 =,for some kn. k ) If = k+ ad i = for some kn, the by lemma., D C 1, i 1) =, D C, i 1), D C3, i 1), D C4, i 1) ad D C, i 1). iv) ) Sice D C, i 1) =,by lemma., i 1 or i 1. Sice D C, i 1), by lemma., + 1 i 1. Therefore i 1 possible. Therefore i 1. Therefore i -4.Therefore i -3. Sice D C3, i 1) =, i-1-4. Therefore i -3. Hece i = 3. ) if i = 3 the by lemma., D C 1, i 1), D C, i 1), D C3, i 1) =, D C4, i 1) ad D C, i 1) =. v) ) Sice D C 1, i 1), D C, i 1), D C3, i 1), D C4, i 1) 1 ad D C, i 1), the by applyig lemma., i 1 1, i 1, 3 i 1 3), 3 i 1 4) ad 3 i 1. So 1 i 1 ) ad hece i 4. 1 ) If + 1 i 4, the the result follows from lemma.. is ot Theorem.7 For every 6 ad i, i) If D C 1, i 1) = D C, i 1) = D C3, i 1) = D C4, i 1) = ad D C, i 1) the D C, i ) = {1, 6,.. 4 },{, 7,.. 3 }, {3, 8,.. },{4, 9,.. 1 },{, 10,.. }, ii) If D C, i 1) = D C3, i 1) = D C4, i 1) = D C, i 1) = ad D C 1, i 1) the D C, i ) = [], 7 Page

5 Domiatig Sets Ad Domiatio Polyomials Of Square Of Cycles iii) If D C 1, i 1) =, D C, i 1), D C3, i 1), D C4, i 1) ad D C, i 1) the D C, i ) = {1, 6,.. 4 },{, 7,.. 3 },{3, 8, }, {4, 9 1 },{, 10,.. }, X 3.X 4 1 if X X, X 3 D C3, i 1), X 4 D C 4, i 3 if X 1) iv) If D C3, i 1) =, D C, i 1), D C, i ) = [] - {x}/x[].,, X D C, i 1) }.1) D C 1, i 1), the v) If D C 1, i 1), D C, i 1), D C3, i 1), D C4, i 1), D C, i 1) the D C, i ) = X 1 {}, X { - 1}, X 3 {-}, X 4 { - 3}, X {-4}/ X 1 D C 1, i 1), X D C, i 1), X 3 D C3, i 1), X 4 D C4, i 1), X D C, i 1).....) i) Sice, D C 1, i 1) = D C, i 1) = D C3, i 1) = D C4, i 1) = ad D C, i 1) by lemma.6) i) = k, ad i=k for some kn. The set {1, 6,.. 4 },{, 7,.. 3 },{3, 8,.. }, {4, 9,.. 1 },{, 10,.. }, has elemets ad it covers all vertices. The other sets with cardiality {1, 6,.. },{, 7,.. },{3, 8,.. }, {4, 9,.. },{, 10,.. }, Clearly, all these are ot domiatig sets. Therefore {1, 6,.. 4 }, {, 7,.. 3 },{3, 8,.. }, {4, 9,.. 1 },{, 10,.. }, is the oly domiatig set of cardiality = k = i. ii) We have D C, i 1) =, D C3, i 1) = D C4, i 1) = ad D C 1, i 1). By lemma.6) ii). We have i =. So,, ) iii) We have D C 1, i 1) =, D C, i 1), D C, i 1). k By lemma.6). iii), = k+ ad i = = k+1 for some kn. D C i = []. D C3, i 1), D C, i 1) = D C4, i 1) ad We deote the families Y 1 = {1, 6,,k-4,k+1 },{, 7,, k-3,k+ }, {3, 8, k-,k+3},{4, 9 k-1, k+4 },{, 10, k,k+ } ad 8 Page

6 k if X Y = X 3 k.x 4 k 1, X k 1 if X X 4 D C4, i 1), X D C, i 1) We shall prove that Domiatig Sets Ad Domiatio Polyomials Of Square Of Cycles D Ck, k 1) = Y 1 Y.Sice k, X 3 D C3, i 1), D C, k ) = {1, 6,,k-4,k+1 }, {, 7,, k-3,k+ },{3, 8, k-,k+3},{4, 9 k-1, k+4 },{, 10, k,k+ },the Y D C, k 1).Also it is obvious that Y D C, k 1). 1 k k Therefore Y Y D C, k 1)..3) 1 k ) Now let Y D Ck, k 1). The k +, k+1 or k is i Y. If k+ Y, the by lemma 3), atleast oe vertex labeled 1, or 3 ad k+1, k or k-1 is i Y. If k+1 or k is i Y, the Y {k+} D Ck 1, k), a cotradictio; because D Ck 1, k) =. Hece k 1 Y, k Y ad k + 1 Y. Therefore, Y = X {k+} for some X D Ck, k ).Hece Y Y1. Now suppose that k + 1 Y ad k+ Y. BY lemma.3) at least oe vertex labeled k, k 1or k is i Y. If ky the Y {k+1} D Ck, k ) = {1, 6,,k-4,k+1 },{, 7,, k-3,k+ },{3, 8, k-,k+3},{4, 9 k-1, k+4 },{, 10, k,k+ } a cotradictio because k X for all X D C, k ). k Therefore k 1, or k is i Y, but ky. Thus Y= X {k+1} for some of X D Ck 1, k). So D Ck, k 1) Y 1 Y....4) From.3) ad.4), D C, i ) = {1, 6,.. 4 },{, 7,.. 3 },{3, 8, }, {4, 9 1 },{, 10,.. }, X 3.X 4 1 if X X, X 3 D C3, i 1), X 4 D C 4, i 3 if X 1),, X D C, i 1) } iv) If D C3, i 1) =, D C, i 1), D C 1, i 1), by lemma.6) iv) i = 3. Therefore D P, i ) = {[] - {x} / x[]}. For example DP,) = {[6] - {x} / x[6]}. 6 {1,,3,4,,6}-{1,,3,4,,6} /6[6]}. = {1,,3,4, }-{{1,,3,4,6}-{{1,,4,,6}-{{1, 3,4,,6}-{{,3,4,,6} v) D C 1, i 1), D C, i 1), D C3, i 1), D C4, i 1), D C, i 1). Let X 1 D C 1, i 1),so atleast oe vertex labeled -1, or 3 is i X 1. If -1, - or -3 X 1, the X 1 {} D C, i ). Let X D C, i 1), the or 3 or 4 is i X. If -, -3 or -4 X, the X { 1} = D C, i ). Now let X 3 D C3, i 1) the 3, 4 or is i X 3. If 3 or 4 or X 3,the X 3 { } = D C, i ). Now let X 4 D C4, i 1), the 4, or or 6 is i X 4. If -4 X 4,the X 4 {-3} D C, i ), for X {, -1}. If - X,the X {-4} D C, i ). 9 Page

7 Domiatig Sets Ad Domiatio Polyomials Of Square Of Cycles Thus we have X 1 {}, X { - 1}, X 3 {-}, X 4 { - 3}, X {-4}/ X 1 D C 1, i 1), X D C, i 1), X 3 D C3, i 1), X 4 D C4, i 1), X D C, i 1) D C, i )...) Now suppose that -1 Y, Y the by lemma 3), atleast oe vertex labeled -, -3 or i Y. If - Y the Y=X 3 ) {-} for some, X 3 D C3, i 1). Now suppose the, - Y & -4 Y, the by lemma 3) oe vertex labeled -6, -7, i Y. If -4Y, the Y = X ) {-1} for X, D C, i 1), So D C, i ) X 1 {}, X { - 1}, X 3 {-}, X 4 { - 3}, X {-4}/ X 1 D C 1, i 1), X D C, i 1), X 3 D C3, i 1), X 4 D C4, i 1), X D C, i 1)....6) From.) &.6) D C, i ) = X 1 {}, X { - 1}, X 3 {-}, X 4 { - 3}, X {-4}/ X 1 D C 1, i 1), X D C, i 1), X 3 D C3, i 1), X 4 D C4, i 1), X D C, i 1). III. Domiatio Polyomial Of A Square Cycle C ) i Let D C, x ) = d C, i) x be the domiatio polyomial of a cycle C. I this sectio, we study this polyomial. i Theorem 3.1 i) If D C, i ) is the family of domiatig sets with cardiality i of C, the d C, i) d C, i 1) d C, i 1) d C, i 1) d C, i 1) d C, i 1) ii) For every 6, D C, x ) = x D C 1, x) D C, x) D C3, x) D C4, x) D C, x) 3 D C, x) x, D C, x) x x, D C, x) x 3x 3 x, 1 3 D C, x) x 4x 6x 4 x, D C, x) x x 10x 10x x with the iitial values i) It follows from theorem.7 ii) It follows from part i) ad the defiitio of the domiatio polyomial. Usig Theorem 3.1, we obtai d C, i ) for 1 1 as show i table 1. Table 1: d C, i ), the umber of domiatig set of C with cardiality i. i Page

8 Domiatig Sets Ad Domiatio Polyomials Of Square Of Cycles I the followig Theorem, we obtai some properties of d C, i ). Theorem 3. The followig properties hold for the coefficiets of D C, x ) ; i) ii) iii) iv) v) vi) vii) i) d C, ) for every N., ) 1, 1), ), 3), 4) d C for every N d C for every d C C for every 3 3 d C C for every 4 d C C for every 4 d P i, ) for i = 1,.., = 1,. D C, ) = {1, 6,.. 4 },{, 7,.. 3 },{3, 8,.. }, {4, 9 1}, {, 10 }, we have d C, ). ii) Sice D C, ) i = [],we have the result. iii) Sice D P, i ) = {[] - {x} /x[]}, we have d C, 1). iv) By iductio o The result is true for = 3 L.H.S. = dc,1) = 3 from table) 3 3. R.H.S = = 3 Therefore, the result is time for = 3 Now suppose that the result is true for all umbers less tha ad we prove it for. By theorem 3.1), d C, ) d C, 3) d C, 3) d C, 3) d C, 3) d C, 3) ) ) = ) = = 1) =. v) By iductio o The result is true for = 4 L.H.S = dc,1) = 4 from table) 4 R.H.S = dc 4,1) = Page

9 Domiatig Sets Ad Domiatio Polyomials Of Square Of Cycles Therefore the result is true for = 1 Now suppose the result is for all atural umbers less tha. By theorem 3.1, d C, 3) d C, 4) d C, 4) d C, 4) d C, 4) d C, 4) = 1) ) 3) ) 3) 3) 1 6 = 1 1) ) 3) 3 ) 3) 6 3) 6 6 = 1 1) ) 3) 3 ) 3) 6 3 1) 6 = 1 1) ) 3) 3 ) 3) 6 ) 6 = 1 1) ) 3) 3 ) 3 ) 6. = 1 1) ) 3) 3 ) 1) 6 = 1 1) ) 3 3) 6 1 = 1) ) 6 vi) By iductio o Let = L.H.S = dc,1) = from table) R.H.S = 1) ) 3) = = Therefore the result is true for =1 Now suppose that the result is true for all atural umbers less tha or equal to. d C, 4) d C, ) d C, ) d C, ) d C, ) d C, ) ) ) 3) 4) ) 3) 4) 3) 4) = 4) = 1 1) ) 3) 4) 4 ) 3) 4) 1 3) 4) 4 4) 4 4 = 1 1) ) 3) 4) 4 ) 3) 4) 1 3) 4) 4 4 1) 4 = 1 1) ) 3) 4) 4 ) 3) 4) 1 3) 4) 4 3) 4 = 1 1) ) 3) 4) 4 ) 3) 4) 1 3) 4 ) 4 = 1 1) ) 3) 4) 4 ) 3) 4) 1 3) ) 4 = 1 1) ) 3) 4) 4 ) 3) 4 3) Page

10 Domiatig Sets Ad Domiatio Polyomials Of Square Of Cycles = 1 1) ) 3) 4) 4 ) 3) 1) 4 = 1 1) ) 3) 4 4) 4 1 = 1) ) 3) 4 Theorem 3.3 i) i, =, 1 i d C i i ii) For every j d C 1, j 1 d C, j1 d C, j d C, j iii) If S d C, j, the for every 6, S = S -1 + S - + S -3 + S -4 + S - with iitial values i S 1 =1, S =3, S 3 =7,S 4 =1 ad S = 31 i) by iductio o First suppose that = the, 10 d C i i d C i i,) 7,1 k d C, k ) k 1, 1 k, 1 k 3, 1 k i d C k d C k d C k d i i i C i 4, k 1 ik ik ik ik ik k d C ik i, k 1 k 1) k 1) 1) 1) 1 k 3 k d Ci k d Ci k d Ci k d Ci4 k,,,, ik1 ik1 ik1 ik1 k1) d C i k 1 i k = k d C i, k 1, ik1 We have the result. ii) By Theorem 3.1, We have d C 1, j 1) d C, j 1) = d C, j) d C, j) d C, j) d C, j) d C, j) d C, j) d C, j) d C 1, j) d C, j) d C3, j) d C4, j) d C 1, j) d C, j) d C, j) d C, j) d C, j) d C, j) d C, j 1) d C, j 1) d C, j) d C, j) 1 Therefore we have the result iii) By theorem 3.1, we have 13 Page

11 i S d C, j Domiatig Sets Ad Domiatio Polyomials Of Square Of Cycles, 1, 1), 1), 1), 1) i S d C j d C j d C j d C j d C j 1, , 1 d C, 1 j d C j d C3 j i 1 i 1 i 1 1 +, 1 1 d C, 1 4 j d C j i 1 i 1 S = S -1 + S - + S -3 + S -4 + S - Refereces [1]. S.Alikhai ad Y.H.Peg, Itroductio to domiatio polyomial of a graph. arxiv:090.1v1[math.co] 14 May 009. []. S.Alikhai ad Y.H.Peg, 009, Domiatio sets ad Domiatio Polyomials of paths, Iteratioal joural of Mathematics ad Mathematical Scieces. Article ID [3]. G.Chartad ad P.Zhag, Itroductio to Graph Theory, McGraw-Hill, Bosto, Mass, USA, 00. [4]. T.W.Hayes,S.T.hedetiemi,ad P.J.Slater, Fudametal of Domiatio i graphs,vol.08 of Moographs ad Textbooks i Pure ad Applied Mathematics, Marcel Dekker, New York,NY,USA,1998. []. S.Alikhai ad Y.H.Peg, Domiatio sets ad Domiatio Polyomials of cycles, arxiv: v [math.co] 0 May 009. [6]. A.Vijaya ad K.Lal Gipso, Domiatio sets ad Domiatio Polyomials of Square paths, accepted i Ope Access joural of Discrete Mathematics. USA Page