An Algorithm for Odd Graceful Labeling of the Union of Paths and Cycles

             !" !# $ "% #% 10.5121 /jgraphhoc.2010.2108 112                                                              M. Ibrahim Mouss a Faculty of Comp uters & Information, Benha University, Benha , Egypt moussa_6 060@yahoo. com ABSTRA CT In 1991, Gnanajot hi [4] proved that the path g raph n P with n vertex and 1 n − edge is odd graceful, a nd the cyc le graph m C with m vertex a nd m edges is o dd gr aceful if an d only if m even, she proved the cycle graph is not gr aceful if m odd. In t his pa per, firstly, we s tudied the g raph m n C P ∪ when 4, 6, 8, 1 0 m = and then we proved that the graph m n C P ∪ is odd graceful if m is even. Finally, we describe d an algorithm to label the v ertices and t he edges of the ve rtex set ( ) m n V C P ∪ and the edge set ( ) m n E C P ∪ . KEY WORDS Vertex label ing, edge la beling, odd gracef ul, Algorit hms 1. INTRODU CTION The study of graceful graph s and graceful labeling methods was introduced by Rosa [1]. Rosa defined a  - valuation o f a graph G with q edges an injection from t he vertices of G to t he s et { } 0 ,1 , 2 q … such that when ea ch edge uv is assigned th e l abel ( ) ( ) , f u f v − the resulting edges are distinct.  - Valuation is a f unction that produces graceful labeling. However, the term graceful labeling was not used until Golomb studied such l abeling several years later [ 2]. A graph  G  of  size  q  is odd-graceful, if there is an inj ec tion f from  ( ) V G  t o { } 0 ,1 , 2 2 1 q … − such that, when each edge uv is assigned the label or weight ( ) ( ) , f u f v − the resu lting edge labels are { } 1 , 3 , 5 2 1 . q … −  This definition was in troduce d by Gnanajot hi [4] in 19 91. In 1991 , Gnanajothi [4] prove d the graph 2 C K m × is odd-graceful if and only if m even. She also prov ed that the graph obtained from 2 n P P × by deleting an edge that joins to end points of the n P paths and this last graph knew as the ladder graph. She proved that every graph with an odd cycle i s not odd graceful. She also prove d the follo wing graphs are odd graceful: ; n m P C if and only if m is even  and the disjoint un ion of copies of 4 C . In 2000, Kathiresan [6] used the notation ;m n P to denote the graph obtained b y ide ntifying the end points of m paths each o ne ha s length n . In 1997 Eldergill [5] genera lized Gnanajothi [4] result on st ars by showing that t he graph ob tained b y joining one end poi nt from each of a ny odd number of paths of equal length is od d graceful graph. In 2 002 Sekar [7] p roved that the graphs; ; m n P when 2 n ≥ and m is odd , 2 ; m P and m  2, 4 ; m P and  m  2, ; m n P when n and m are even a nd n  4 and m  4, 4 1 ; 4 2 , r r P + +  4 1 ;4 , r r P −  and all n -pol ygona l snakes with n even ar e odd graceful. In 2009 Mo ussa [9] presented some algorithms to prove for all 2 m ≥ t he followi ng graphs 4 1 ;m 1 ,2 ,3 , = r P r − and              !" !# $ "% #% 113 4 1 ;m =1,2 , r r P + are odd graceful. He pr esented algorit hm which sh ow that the closed spider graph and the graphs obtained b y joining one or two paths m P to each vertex of the path n P are odd graceful. He used the cy cle representa tio n a nd de noted £ -represen tation to prese nt a simple labeling graph algo ri thm, the cy cle representation is similar to the π -representation made by Kot azig [ 3]. In 2009 Moussa and Bader [ 8] have presented t he algorithms t hat show ed the graphs obtained by joining n pendant ed ges t o e ach vertex o f m C are odd graceful if and only if m is even. In t his paper we show that n m C P ∪ is odd grace ful if m is even and 2 n m > − , if t he number of edges in the cyc l e m C can be equally divisible by four , and 4 n m > − for all other even value of m . We first explicitly define an odd graceful labeling  of 4 n C P ∪ , 6 , n C P ∪ 8 1 0 a n d , n n C P C P ∪ ∪ and then, using this o dd grace ful labeling, describe a recurs ive procedure to obtain an odd g raceful labeling o f n m C P ∪ . Finall y; we present an algorith m for comput ing the odd grace ful labeling of the un ion of path and cycle graphs, we pro ve the correctness of the algorithms at the end of this paper. The remainder o f this paper is organized as foll ows. I n section 2 , illustrate the need of graph labeling and we mentio n the existing variety typ es of l abeling methods. In sec tion  we give some assumptions and defin itions r elated with the odd graceful labeling and graphs. In section  , we pr ese nt and discuss t he odd l abe l o f union of paths a nd c ycle. Section  i s the conclusion of this researc h. 2. R ELATED WORK  The graph labeling serves as useful models for a broad ran ge of applications such as: radar, communications network, circuit d esig n, coding th eory, astronomy, x-ray, crystallography, data  base management and models for constraint programming over fi nite domains. J. Gallian i n his dynamic survey [11], he has collected everything on graph labeling, he observed t hat over thousand papers have been studied and many ki nds o f graph labeling have been defi ned, viz.: Graceful Labe ling, Harmonious Labe ling, Magic Labelings, b alanced labeling, k -graceful Labeling,  - labeling, and Odd-Graceful La belings. For further information a bout the graph labeling, we adv ise the reader to r efer to the brillia nt dynamic surve y on the subject [11]. 3. A SSUMPTIONS AND D EFINITIONS . Def inition 1[4] Let G be a f inite simple graph, whose vertex set is denoted ( ) V G , while ( ) E G denotes its edge set, the order of G is th e cardinality ( ) V G n = and the size of G is the cardinality ( ) G E q = . We write ( ) uv E G ∈ if there is an edge connecting the vertices u and v in G . An odd graceful labeling of a graph G is a one to one function : ( ) { 0 ,1 , 2 ,... , 2 1 } . f V G q → −  Su ch that, when each edge  uv is assigned the label ( ) ( ) ( ) f f f u v u v − ∗ = the resulting edge labels are { } 0 , 1 , 2 2 1 q … − . Def inition 2[10] A path in a graph is a sequ ence of vertices such that from each of its verti ce s t here is an edge to the ne xt vertex in t he sequence. Th e first vertex is called the start vertex and the last vert ex is called the end vertex. Bot h of t hem are called end or t erminal vertices of the path. Th e other vertices in t he path are internal vertices. A cycle is a graph with an equal number of vertices and edges whose verti ce s c an be placed around a circle so that two vertices are adjace nt if and only if they appear consecutively along the circle. The graph has  n or  m vertex th at is a path or a cycle is denoted n P or m C , respectively.  The union of t wo graphs ( ) 1 1 1 , G V E =  and ( ) 2 2 2 , G V E = , written 1 2 G G ∪ , is the graph with vertex set 1 2 1 2 ( ) ( ) ( ) V G G G G V V ∪ = ∪ and the edge set 1 2 1 2 ( ) ( ) ( ) E G G G G E E ∪ = ∪ .               !" !# $ "% #% 114 4. V ARIATION OF ODD GRACE FUL LABELING 4 .1. Odd gr acefulness of 4 n C P ∪ Theore m1 4 n C P ∪ is odd graceful for every intege r 2 n > . Proof Let 4 1 2 3 4 ( ) { , , , } V C u u u u = , 1 { } ( ) , ... , n n V v v P = where 4 ( ) V C is the vertex set o f t he cycle 4 C  and  ( ) n V P  is t he vertex set of the path n P , and 3 q n = + , see Fig.1. For every vertex i u and i v , t he odd graceful labeling functions  ( ) i u f and ( ) i v f respectively as follows { 1 2 3 4 2 6 ( ) 0 , ( ) 2 1 , ( ) 2 , ( ) 2 5 a n d ( ) i i o d d i q i i e v e n f u f u q f u f u q f v − − = = − = = − = The edge labeling function * f defined as follows: 1 2 2 3 * * * 3 4 4 1 * 1 ( ) 2 1 , ( ) 2 3 , ( ) 2 7 , ( ) 2 5 , a n d ( ) 2 2 7 1 , 2 , . . . , 1 i i f u q f u q f u q f u q f v v q i i n u u u u ∗ + = − = − = − = − = − − = − Figure1shows the method labeling of the graph 4 n C P ∪ this complete the proof.   24 22 20 18 16 14 0   31 27  23  19  15  11  7  3  31  27  21  17  13  9  5  1 29 25 1 3  5  7  9  11  13  Figure 1. 4.2. Odd grace fulness of 6 n C P ∪ Theore m2 6 n C P ∪ is odd graceful for eve ry integer 2 n > . Proof Let 6 1 2 3 4 5 6 ( ) { , , , } , , V C u u u u u u = , 1 { } ( ) , . . ., n n V v v P = where  6 ( ) V C is the vertex set o f the cycle 6 C  and ( ) n V P is t he v ertex s et of the p ath n P , and 5 q n + = , see Fig. 2. For every vertex i u and i v , the odd graceful labeling function s ( ) i u f and ( ) i v f respective ly as follows: 1 2 3 4 5 6 2 3 1 2 1 2 ( ) 0 , ( ) 2 1 , ( ) 2 3 , ( ) 4 , ( ) 2 9 a n d ( ) ( ) 2 , i i i o d d i i q i i e v e n f u f u q f u q f u f u q f v f u + ≤ = − + = = − = − = = − = =      The edge labeling function * f defined as follows: 1 2 2 3 * * * 3 4 4 5 * * 5 6 6 1 ( ) 2 1 , ( ) 2 3, ( ) 2 5, ( ) 2 7 , ( ) 2 13 , and ( ) 2 9 u u u u u u u u u u u u f q f q f q f q f q f q ∗ = − = − = − = − = − = −              !" !# $ "% #% 115 18  16  14 12  0     27   19 11  27 19 17  13  9  7  5  3 1 25 15 2 4 1  5  7  9  11 23 21 Figure 2. 25  4.3. Odd gracefulness of 8 n C P ∪ Theore m 3 8 n C P ∪ is odd graceful if a nd only if n  7. Proof : For every vertex i u and i v in 8 ( ) n V P C ∪ , we defined the odd g raceful labeling functions ( ) i u f and ( ) i v f respective ly as f ollows ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 6 7 0 , 2 1 , 2 , 2 3 , 4 , 2 5 , 6 , u u q u u q u u q u f f f f f f f = = − = = − = = − = ( ) 8 2 1 4 2 2 1 4 4 2 1 3 , a n d ( ) i i o d d q i q i i e v e n i u q f v f − = − − ≤ = − =      And the function * f is defined as follows: 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 1 * * * * * * * * ( ) 2 1 , ( ) 2 3 , ( ) 2 5 , ( ) 2 7 , ( ) 2 9 , ( ) 2 1 1 , ) 2 1 9 , ( ) 2 1 3 ( ) 2 1 5 , ( ) 2 1 7 , ( ) 2 2 1 5 , 3 , . . . , 8 * * * 1 2 2 3 1 ( u q f u q f u q f u q u q f u q f q f u q f v v q f v v q f v v q i i q i i f u u u u f u u u u u = − = − = − = − = − = − = − = − = − = − = − − = − + The labeling of the gra ph 8 1 2 C P ∪ is indicated by F ig.3.  0 24  2    18      14                37     23  17  13  9  5  1               21  15  11  7  3              1  3  5  7  9  11 35               Figure 3. 4.4. Odd grace fulness of 10 n C P ∪ Theore m 4 10 n C P ∪ is odd graceful for eve ry integer n   . Proof: For every vertex i u and i v , we defi ned the odd graceful labeling functions ( ) i u f and ( ) i v f respectively as follows:              !" !# $ "% #% 116 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 6 7 8 9 1 0 0 , 2 1 , 2, 2 3 , 4 , 2 5, 6 , 2 7 , 8, 2 1 7 , a n d u u q u u q u u q u u q u u q f f f f f f f f f f = = − = = − = = − = = − = = − 2 5 1 , 3 2 1 6 ( ) i i o d d i i q i i e v e n i f v + ≤ = − − =      So we obtain all the ed ge labels and the function * f is defined as follows: 1 2 2 3 3 4 4 5 * * * * ( ) 2 1, ( ) 2 3 , ( ) 2 5 , ( ) 2 7 u u u u u u u u q f q f q f q f = − = − = − = − 4.5. Odd grace fulness of m n C P ∪ Theore m 5 Let k is a given inte ger and 2 k m = , the graph n m C P ∪ is odd graceful for every 2 n m > − , k is even, if k  is odd number the graph n m C P ∪ is odd grace ful f or every 4 n m > − . Proof: Let 1 ( ) { , ..., } m m V C u u = , 1 ( ) { , ... , } n n V P v v = , where  ( ) m V C is the vertex s et of the cy cle m C and ( ) n V P is the vertex set of t he path n P , and 1 m q n + − = . For e very vertex i u and i v , we defined the odd graceful labeling functions ( ) i u f and ( ) i v f respectively a s follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 6 9 0 , 2 1 , 2 , 2 3 , 4 , 2 5 , 8 , .. . , 2 (2 3 ) m q q q q m u u u u u u u u f f f f f f f f = = − = = − = = − = = − − If the value 2 , k k m =  is odd number, t he vertex 2 v would be labeled 2 2 2 2 ( ) q m f v − + = which decreased by two at every new value 4 , 6 , .. . , 2 k i − = , this means that 2 ( ) ( ) 2 2 2 ( 4 ) i i f v f v q m i − = − = − − − , and 2 1 , 3 , . . . , 2 2 2 ( 4 ) ( ) i i k i o d d i i k q m i i e v e n f v + ≤ = − − − − =      If 2 , k k m = is even n umber, the vert ex 2 v would be labeled 2 2 2 2 ( ) q m f v − + = which d ecreased by two at every new value 4 , 6 , .. ., 2 k i − = . For 2 i k − = the label value is 2 2 2 6 ( ) k q m k f v − − + − = whi le the label value of the vertex k v is four out of the value 2 ( ) k f v − , this means that 2 4 ( ) ( ) 2 2 2 i k k f v f v q m i − = = − = − + − , and 2 2 4 2 , 4 , . . . , - 2 2 2 2 ( ) q m i i i i o d d i k q m i k i e v e n f v − + = − − + − ≤ =                   !" !# $ "% #% 117 The function * f induces the edge labels of the cycle m C as the following: 1 2 2 3 3 4 * * * ( ) 2 1 , ( ) 2 3 , ( ) 2 5 , . . . , u q f u q f u q f u u u = − = − = − 1 1 * * ) 2 3 5 , ( ) 2 2 3 ( m m m f q m f u q m u u u − = − + = − + . Function * f induces the edge labels of the path  as follows: 1 2 2 3 2 1 1 1 * * * * * ( ) 2 2 1 , ( ) 2 2 1, .. ., ( ) 2 3 7 , ( ) 2 3 3 , .. . . .. . , ) 2 3 1, . . . .. . . ., 1 ( k k k k k k v q m f v q m v q m f v q m f q m f v v v v v v f − − − + = − + = − − = − + = − + = − + There is a guarantee that each component in the given graph has odd graceful, th e path graph is odd graceful, the cycle graph wi th an e ven number of vertices is odd graceful (see [4]) . We have t o pro ve that the vertex l abels are di stinct and all the ed ge lab els are disti nct od d numbers { } 1 , 3 , 5 2 1 q … − . T he edge labels of m C are n umbered a ccording to t he d ecreas ing sequence 2 1, 2 3, ..... . q q − − . The edge labels o f n P are numbered according to the decreasing sequence * 1 ( ) 2 2 (2 1 ) , 4, .. . , i i f v v q i m i q m + = − − − = − . The reader can easily f ind out, if i q m = − the l as t edge label is equal o ne; this means t hat the edge labels t ake values in { } 2 1 , 2 3 , , 1 q q − − … . In o rder to prevent any vertex i n n P to share label with a vertex in m C , the difference between th e largest even label and the smallest even label in n P have to b e more t han the largest even label in m C , this leads to two cases: Case I: if 2 , m k k = is even then 2 1 ) 2 2 ( ( ) 2 m n f u m n m − − > −  > −     Case I: if 2 , m k k = is odd then 2 1 ) 2( 2 ( ( ) 1 ) 4 m n f u m n m − − − > −  > −     .  4.6. T he proposed sequential algorithm The u nion graph  n m P C ∪  has a vertex set ( ( ) ( ) ) n m n m V P V P V C C ∪ = ∪ with cardinality n m + and an edge set ( ( ) ( ) ) n m n m E P E P E C C ∪ = ∪ with cardinality 1 q m n = + − .  Let the cycle m C is demonstrated by listing the vertices and t he edges in t he order 1 1 2 2 1 1 1 , , , , .. ., , , , , m m m m u e u e u e u e u − − . We name the vertex m u ACTIVE vertex, the vertex m u is an endpoint of the edge 1 m e − , and we name the edge 1 m e − DOUBLE-JUMP edge . The path n P is demonstrated by listing the vertices and the edges in the o rder 1 1 2 2 1 1 , , , , . . . , , , n n n v e v e v e v − − ′ ′ ′ ′ . The algorithm has two passes; t hey c an run in a sequential or a parallel way and it work s in a way similar to the above labeling in section 3.5. In one pass, t he algorithm labels th e vertices and the edges in the cycle m C . For the other pass, it labels the verti ces and t he edges of the path n P . At the beginning of the algorithm, we are computing th e o dd label function f or the ACTIVE vertex and the DOUBLE-JUMP edge. The ACTIVE vertex has the odd graceful labeling function ( ) 2 (2 3 ) m f u q m = − − , t he vertex m u ha s the smallest odd label value b etween the vertices in the c yc le . m C The DOUBLE -JUMP edge i s assigned the label fun ction * 1 ( ) 2 3 5 m f e q m − = − + . The given label to the ACTI VE vertex and the DOUBLE-JUMP edge computed independently from other vertices or edges in the graph. 1. N umber the ACTIVE vertex with the value ( ) 2 ( 2 3) m f u q m = − − 2. N umber the DOUBLE-JUMP edge with the v alue * 1 ( ) 2 3 5 m f e q m − = − + Algorithm 1: Procedure Initialization In the f irst pass, the algorit hm starts at the vertex 1 u , there are two main steps that can b e performed. These st eps (in particular order) ar e: performing an action o n the current vertex              !" !# $ "% #% 118 (referred to as " numbering" the vertex), number the current vertex wit h the value ( ) 0 1 f u = , traversing to the l eft adjac ent vert ex 2 u and number i t with the value ( ) 2 1 2 q f u = − , and traversing to t he left adjacent vertex 3 u and number it wi th the value ( ) 3 2 f u = traversing to the left ad jacent v ertex 4 u and number it with the value ( ) 2 3 4 q f u = − , tr av ersing to the l eft adjacent vertex 5 u and numbe r it with the value ( ) 5 4 f u = .. Thus t he process is most easily described through recursio n. Finally, reach to the ACTIVE vert ex which h as the exceptio n  label and number it with the value ( ) 2 (2 3 ) m f u q m = − − , the edge’s labeling induced by the absolute value of the di fference of the vertex’s labeling. To la bel t he cy cle m C odd graceful, perfor m the followi ng opera tio ns, starti ng with 1 u : 1. Nu mbe r the vertex 1 u with the value ( ) 0 1 f u = 2. For ( i = 3; i  m-2; i += 2 ) ( ) ( ) 2 2 i i f u f u − = + 3. F or ( i = 2; i  m-1; i += 2 ) ( ) 1 2 i f u q i = − + 4. Number the ACTIVE vertex with ( ) 2 (2 3 ). m f u q m = − − 5. Compute the edge labels by taking the absolute value of the difference of incident vertex labels. Algorithm2: Odd graceful labeling of m C After t he above process, the algorithm starts t he second pass to l abel the v ertices and edges of the path component n P . Second p ass starts at the edge 1 2 1 ( , ) e v v ′ = , it s label value is 1 2 ( ) ( ) m f e f u ∗ − ′ = , if the label value of the edge 1 e ′ equals to th e label value of the DOUBLE- JUMP ed ge renumber it wit h the value ( ) ( ) 2 1 1 f f e e = − ∗ ∗ ′ ′ and number the vertex 1 v wi th the label value 1 ( ) 1 , f v = trave rsing to the vertex 2 v and nu mber it with the value 2 1 ( ) ( ) 1 f e f v ∗ ′ = + . Traversing to the next incid ent edge 2 e ′ and number i t with the value ( ) ( ) 2 2 1 f f e e = − ∗ ∗ ′ ′ if the l abel value of the edge 2 e ′ equals to the label value of the DOUBLE-JUMP edge re number it w ith the value ( ) ( ) 2 2 2 f f e e = − ∗ ∗ ′ ′ , traverse to the next vertex 3 v which in duces the label value 3 2 2 ( ) ( ) ( ) f v f v f e ∗ = ′ − , otherwise tra verse to the ne xt vertex 3 v , without double subtracting for the label value of the edge 2 e ′ , and number it wit h the value 3 2 2 ( ) ( ) ( ) f v f v f e ∗ = ′ − , trave rse to the next ve rte x 4 v which in duce s the l abel va lue 4 3 3 ( ) ( ) ( ) f v f v f e ∗ = ′ + . Thus the process is mos t easily described through recursion again. To label the path n P odd graceful labeling, perform the following operati ons, starting with the edge 1 2 1 ( , ) e v v ′ = : 1. Nu mbe r the vertex 1 v with the value ( ) 1 1 f v = 2. Nu mbe r an auxiliary edge 0 e ′ with 0 ( ) ( ) m f e f u ∗ ′ = 3. F or ( j = 1 ; j  n- 1; j += 1 ) 3.1 Number the edge j e ′ with ( ) ( ) 2 1 j j f f e e = − ∗ ∗ − ′ ′ 3.2 If ( ( ) ( ) 1 m j f f e e − = ∗ ∗ ′ )  Renumber the edge j e ′ with the value ( ) ( ) 2 j j f f e e = − ∗ ∗ ′ ′ 3.3 Number the vertex 1 j v + with the value 1 1 ) ( ) ( ( ) ( ) j j j j e f v f f v + + ∗ − ′ + = 4. E nd For              !" !# $ "% #% 119 Algorithm 3: Odd graceful labeling of n P The algorithm is traversed exactly once for each vertex and edge in the graph n m P C ∪ , since the size of the graph equal s q then at most O ( q ) time is spen t in total labeling of the vertices and edges, th us the total ru nning time of th e algorithm is O ( q ). The parallel algorithm f or the odd graceful labeling o f the graph n m P C ∪ , b ased on the above proposed sequential alg orit hm is building easily. Since all the above thr ee subroutin e are independ ent and t here i s no rea son t o sort their executing out, so they are to join up parallel in the same time point. 5. C ONCLUSION In this paper, we first explicitl y defined an od d graceful labeling of 4 n C P ∪ , 6 , n C P ∪ 8 1 0 a n d , n n C P C P ∪ ∪ a nd then usin g t his odd graceful labeling to have generalized results b y describing a recursive procedure t o o btain an odd graceful labeling of n m C P ∪ , if m is even and 2 n m > − , if t he number of edges in the cy cle m C can be equally divisible b y f our, and 4 n m > − for all other even value of m . After we in troduced a general form for labeling the union of the paths and t he cycles i n odd graceful l abel, we d escribed a sequential algori thm to label the vertices and the edges of the graph n m P C ∪ . The sequential algorithm runs in l inear with total running time equals O ( q ). The parallel version of th e proposed algor ithm, as w e showed, existed and it i s des cribed shortly.  R EFERENCES [1] A. R osa ( 1967), On certa in valuations of the vertices o f a grap h, The ory of Graphs (Inter net Symposium, R ome, Jul y 1966) , Gordon and Breach, N. Y. and Dunod Pa ris 349-355. [2] S . W. G olomb ( 1972) H o w t o numbe r a gra ph: graph theory a nd comp uting, R. C.Read, ed.Academic P ress: 23-37. [3] A. Kotazig, On cer tain verte x valuation of fin ite graphs, Utilit as Math.4 (19 73)261-290. [4] R.B. G nanajothi (1991), Topi cs in Graph The ory, P h.D. Thesis, Madurai K amara j U niversit y , India. [5] P. Eldergill, Dec omposition of the Complete Graph with an Eve n N umber of V ertices, M. Sc. Thesis, McMa ster Universit y, 1997. [6] K. Kathiresan, Two classes of graceful graph s, Ars. Com bin.55 (2000) 129-132. [7] C. Sekar, St udies in Graph The ory, Ph. D. T hesis, Madurai Kam araj Universi ty, 2002. [8] M. I. M oussa & E. M. 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