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DOI 10.4010/2016.1925 ISSN 2321 3361 © 2016 IJESC Research Article Volume 6 Issue No. 7 Some Domination Parameters of Prime Square Dominating Graphs A. Sudhakaraiah *, V. Raghava Lashmi1, B. Narayana2 Department of Mathematics, S. V. University, Tirupati, Andhra Pradesh, India* Department of S&H (Mathematics), Vijaya Institute of Technology for Women, Vijayawada, A. P., India1 Department of Basic Science and Humanities, St, Mary’s Women’s Engineering College, Guntur, A. P. India2 Abstract The purpose of this paper is to typify Prime Square Dominating Graphs and to discuss some of their domination parameters. Theorems are proposed to establish domination number and complementary tree domination number of some of the Prime Square Dominating Graphs. Moreover, conditions sufficient for Prime Square Dominating Graph characterization of a certain class of graphs are put forth. Keywords: Complementary Tree Dominating Set, Complementary Tree Domination Number, Dominating Set, Prime Square Dominating Graph. I. INTRODUCTION Graph theory is one of the most beautiful branches of modern mathematics with a wide range of research scope. Many real life situations can be modeled as a graph theory problem and can be represented diagrammatically with a set of points called as vertices joined together with lines called as edges. For example, in network problems, network places can be represented by vertices and network links can be represented by edges, so that real life situations can be understood in a better demeanor. Some of the important areas of research in graph theory are domination and the graph labeling. The domination problem in a graph is to find a minimum sized vertex set D such that every vertex not in D is adjacent to at least one vertex in D. The domination number of the graph G is denoted by γ (G) and is the minimum cardinality of a dominating set of G. Dominating sets play an imperative role in algorithms and in combinatorics. Complementary tree dominating set problem is the problem of finding complementary tree dominating set of specific size for the given graph. A dominating set S V of a graph G with vertex set V (G) and edge set E (G) is a complementary tree dominating set if the induced sub graph ˂ V – S > is a tree. Complementary tree domination number is the minimum cardinality of a complementary tree dominating set of G. It is denoted by (G). The notion of complementary tree dominating set is due to S. Muttamai et al. [1]. Some results pertaining to the bounds of Complementary tree domination number are obtained by them. In [2] and [3] Complementary tree domination number of some specified class of interval graphs and circular arc graphs are obtained. Labeling of a graph plays a vital role in graph theory. Seod M.A.and Youssef M.Z [4] and Joseph A. Gallian [5] deal with graph labeling besides many others. A labeling of a graph G is an assignment of distinct positive integers to its vertices. A graph is a prime square dominating graph, if the vertices of graph G are labeled with positive integers such that the vertex labeled with composite number c is adjacent to the vertex named with prime number p if and only if p2/c (p2 divides c). In this paper, the study of complementary tree domination number of International Journal of Engineering Science and Computing, July 2016 graphs is extended to a special class of graphs namely, prime square dominating graphs. Unless otherwise specified, throughout the paper the graph G(V, E) is a finite and connected graph with the vertex set V(G) and edge set E(G) and standard definitions of graph theory as found in [6] are followed. II. IMPORTANT RESUITS Result 2.1: For a connected graph G of order (G) ≤ k-1. k ≥ 2, Result 2.2: For a connected graph G of order (G) ≤ (G). k ≥ 2, γ Result 2.3: For a connected graph, every pendant vertex is a member of all ctd – sets [1]. Result 2.4: Every prime square dominating graph is a finite and simple graph. Result 2.5: Every prime square dominating graph is a prime dominating graph. The converse is not true. Result 2.5: Every graph is a bipartite graph if and only if the graph is a prime square dominating graph [7]. III. DISCUSSIONS CHARECTERIZATION OF GRAPHS AS PRIME SQUARE DOMINATING GRAPHS 3. 1 THEOREM: Every cycle graph with even number of vertices is a prime square dominating graph. Proof: Let be a cycle graph with vertex set V= { , , ,..., }. Label the vertices , ,..., with the prime numbers , , ,..., respectively and the vertices , , ,..., with composite numbers , , ,..., respectively, so that divides , divides , divides ……….. and divides & . 8263 http://ijesc.org/ From the labeling of the graph, it is clear that the vertices of graph are labeled with positive integers such that the vertex labeled with composite number c is adjacent to the vertex labeled with prime number p if p2/c. By the definition of prime square dominating graph, it follows that the graph is a prime square dominating graph. Therefore, the vertices of every cycle graph with even number of vertices can be labeled in such a way that the graph becomes a prime square dominating graph. Hence the result follows. 2 225 2 _ 1 5 Note: A cycle graph with odd number of vertices is a not a prime square dominating graph [7]. 3 Experimental problem: Consider a cycle graph which will be as follows: Prime square dominating graph, Cycle graph Label the odd vertices , of the graph with prime numbers 2, 3, 5 and the even vertices , , with composite numbers 36, 225 & 100 respectively. Clearly, vertex labeled with prime number 2 is adjacent to vertices labeled with composite numbers 100 & 36 and moreover 22 divides36 & 100. Similarly, 3 is adjacent to vertices labeled with composite numbers 225 & 36 and moreover 32 divides 225 & 36 and 5 is adjacent to vertices labeled with composite numbers 225 & 100 and moreover 52 divides 225 &100. Implies, G is a prime square dominating graph. Therefore, labeling of the graph is done in such a way that the graph G is a prime square dominating graph. International Journal of Engineering Science and Computing, July 2016 3.2 THEOREM: Every tree is a prime square dominating graph. Proof: Let T be a tree of height ‘h’. Here two cases will arise. Case i: h may be even. Partition the vertex set into two subsets Such that = set of all vertices at level o, 2, 4, ……, h and = set of all vertices at level 1, 3, 5, ……, h-1 Case ii: h may be odd. Partition the vertex set V into two subsets Such that = set of all vertices at level o, 2, 4, ……, h-1 and = set of all vertices at level 1, 3, 5, ……, h. In both the cases, the vertex set V of the graph is partitioned into two sets and so that every edge of the graph has one end vertex in and another end vertex in Label all the vertices in ith prime numbers and all the vertices in with composite numbers depending upon the labeling of the vertices in such that the vertex labeled with composite number ‘c’ is adjacent to the vertex labeled with prime number ‘p’ if and only if p2/c. By the definition of prime square dominating graph, the tree T becomes a prime square dominating graph. It follows that, every tree is a prime square dominating graph. Experimental problem: Consider a tree as follows: 8264 http://ijesc.org/ Tree with height 3 Partition the vertex set into two subsets that = set of all vertices at level o & 2 and 7 of all vertices at level 1 & 3. Implies { and { Such = set = Prime Square Dominating Graph = Label the vertices of the tree ‘T’ with prime numbers 11, 3, 5, 2 & 7 respectively and the vertices with composite numbers 27225, 23716, 18, 27, 8, 12, 49 & 98 respectively. Clearly vertex labeled with composite number 27225 is adjacent to vertices labeled with prime numbers 11, 3 & 5 and moreover 112, 32 & 52 divide 27225, vertex labeled with composite number 23716 is adjacent to vertices labeled with prime numbers 11, 2 & 7 and moreover 112, 22 & 72 divide 23716. Similarly, 18 & 27 are adjacent to the vertex labeled with prime number 3 and 32 divides 18 & 27; 8 & 12 are adjacent to the vertex labeled with prime number 2 and 22 divides 8 & 12 and 49 & 98 are adjacent to the vertex labeled with prime number 7 and 72 divides 49 & 98. From the definition of the prime square dominating graph, it follows that T is a prime square dominating graph Therefore, labeling of every Tree can done in such a way that the tree becomes a prime square dominating graph. In this case, the corresponding prime square dominating graph will be as follows International Journal of Engineering Science and Computing, July 2016 Domination Number and Complementary Tree Domination number 0f Prime Square Dominating Graphs 3. 3 THEOREM: Let G be a prime square dominating graph with vertex set V= { , , ,…, }, n ≥ 3. Let the vertices with even suffixes be labeled with prime numbers , ….respectively starting with and the vertices with odd suffixes be labeled with composite numbers , …………… respectively starting with such that divides only & for 1≤ i ≤ n and for even values of i.. Then γ (G) = n/2 if n is even and if n is odd and (G) = n-2 Proof: Let the vertex set V={ , , ,…, } of the graph G be labeled with composite numbers and prime numbers satisfying the conditions mentioned in the theorem. It is supposed that divides only & for 1≤ i ≤ n and for even values of i. It follows that divides only & divides only & divides only & …………………… divides only when n is even and divides only & when n is odd 8265 http://ijesc.org/ As the graph is a prime square dominating graph, the edges of the graph are { , }, { , }, { , , { , }, ………………..., { , ,{ , }, { , if n is even and { , }, { , }, { , , { , }, ………….., { , ,{ , }, { , , { , } if n is odd. In both the cases all the edges are distinct and they form a path graph. Hence, γ (G) = n/2 if n is even and if n is odd and (G) = n-2 [1]. Experimental problem: Consider the prime square dominating graph G with vertex set V= { 9, 3, 36, 2, 100, 5, 1225, 7}. It will be as follows: not adjacent to any other vertex, except for k = 2, 3, 4, ……, n. Implies d( ) = 1 for k = 2, 3, 4, ……, n. The vertices , ,... ….., are pendant vertices. Every ctd – 18 set of a graph consists of every pendant23716 vertex of the graph. Every ctd – set of graph G consists of , ,... ………, . Implies, (G) ≥ n – 1. But for any graph of order n≥2, (G) ≤ n – 1. It follows that, (G) = n – 1. The minimum , ,... ….., . As vertex 12 ctd – set is is adjacent to k = 2, 3, 4,……, n, inates 27 all the remaining vertices of the graph. Therefore, the set { 98} 27225 is the dominating set of the graph and it is a minimal dominating set of the graph. Hence, γ (G) = 1 8 9 3 2 5 Experimental problem: Consider the prime square dominating graph G with vertex set V = { 2, 8, 16, 40, 60, }. It will be as follows 36 6 100 1225 8 7 16 Prime Square Dominating Graph Here the vertex labeling satisfies all the conditions mentioned in the theorem, where , , . Also, the order of the graph, n = 8( even ). Hence, γ (G) = 8/2 = 4 and (G) = 9 - 2 = 7. 3. 4 THEOREM: Let the vertex set V = { , , ,..., }, n ≥ 2 of a prime square dominating graph G be such that a vertex , 1≤ i ≤ n is labeled with a prime number and the remaining all vertices are labeled with composite numbers , ,... , such that divides all the vertices labeled with composite numbers. Then G is a prime square dominating graph with γ(G)=1 and (G)=n1. Proof: Let the vertex set V={ , , ,…, } of the prime square dominating graph G be labeled with composite numbers and prime numbers satisfying the conditions mentioned in the theorem. Implies, vertex is adjacent to k = 2, 3, 4, ……, n. As there is no other vertex labeled with prime, except is International Journal of Engineering Science and Computing, July 2016 2 40 60 Prime Square Dominating Graph The graph satisfies all the conditions stated in the above theorem for =2, , , , and moreover order of the graph, n = 5. Clearly, γ (G) =1 and (G) = 4 where Minimal dominating set = { 2 } & minimal ctd – set is {8, 16, 40, 60 } 3. 5 THEOREM: Let the vertex set V = { , , ,..., }, n ≥ 4 of a graph be such that the vertices & 1≤ i, j ≤ n are labeled with a 8266 http://ijesc.org/ prime number and composite number and some of the remaining vertices are labeled with composite numbers and some with prime numbers so that and all the squares of primes divide only . Then γ (G) = 2 and (G) = n-2. vertices labeled with 3, 5, 7, 44, 40, 16, 8 are pendant vertices and the vertices = 2 & = 44100 satisfy all the conditions mentioned in the theorem. Hence γ (G) = 2 & (G) = n – 2= 9 - 2 = 7, Where Minimal dominating set = {2, 44100} & minimum ctd – set is {3, 5, 7, 44, 40, 16, 8 }. Prime square dominating graph in this case will be as follows Proof: Let the vertex set V = { , , ,………, }, of a prime square dominating graph G be labeled with composite numbers and prime numbers satisfying the conditions mentioned in the theorem. By hypothesis, except , the squares of all the other primes divide only one composite number . So, all the vertices labeled with primes except are adjacent only to and are pendant vertices. Let them be p in number. Moreover, all the composite numbers, except are divisible only by . All vertices labeled with composite numbers other than are pendant vertices. Let them be q in number. is adjacent to all the vertices labeled with composite numbers and is adjacent to all the vertices labeled with prime numbers. Therefore, and are not pendant vertices. The order of the graph is n. Implies p + q + 2 = n and p + q = n-2. G has n - 2 pendant vertices. Every ctd – set consists of every pendant vertex of the graph. Every ctd – set of graph G consists of n - 2 vertices. It follows that (G) ≥ n – 2 Let S = V - { }. As vertex is adjacent to every vertex labeled with composite number and .is adjacent to every vertex labeled with prime integer, S is a dominating set. Moreover, is adjacent to . It follows that the induced subgraph ˂ V – S ˃ is a tree. S is a complementary tree dominating set. Implies (G) ≤ n – 2. Therefore, (G) = n – 2 and the minimal complementary tree dominating set is V - { }. It is obvious that the set { } is the minimal dominating set and Therefore, γ (G) = 2. Experimental problem: Consider the prime square dominating graph with vertex set V = {2, 3, 5, 7, 44, 44100, 40, 16, 8}. Here 22 divides 44, 44100, 40, 16 & 8; 32, 52 & 72 divide only 44100. The International Journal of Engineering Science and Computing, July 2016 44 2 44100 3 40 5 16 7 8 Prime Square Dominating Graph IV. Conclusion Resolving the domination parameters of some special classes of prime square dominating graphs has been the main focus of the paper. Some categorized graphs have been chosen in the process of examination. In future, efforts will be put to find dominator chromatic number and other domination parameters of prime square dominating graphs. V. REFERENCES [1] S. Muthammai, M. Bhanumathi and P. Vidhya, Complimentary tree domination number of a graph, International MathematicalForum,Vol.6, no. 26 (2011), 1273 – 1282. [2] Dr. A. Sudhakaraiah, V. Raghava Lakshmi, V.Rama Latha and T.Venkateswarlu, Complimentary tree domination number of interval graphs, International Journal of Advanced Research in Computer Science and Software Engineering, Vol. 3, No. 6 (2013), 1338-1342. 8267 http://ijesc.org/ [3] Dr. A. Sudhakaraiah, V. Raghava Lakshmi, and T.Venkateswarlu, Complimentary tree domination number of circular- arc graphs, International Journal of Engineering and Innovative Technology (IJEIT), Volume 3, Issue 2, August 2013, pg. 229-232. [4] Seod M.A.and Youssef M.Z., “On prime labeling of graphs”, Congr. Number.141(1999), 203-215. [5] Joseph A. Gallian, “A Dynamic Survey of graph labeling”, the electronic journal of combinatorics #D S 6, 2005. [6] F.Harrary, Graph Theory, Addition - Wesley Reading Mass, 1969. [7] A. Sudhakaraiah and B. Narayana, A note on prime square dominating graph, , IOSR Journal of Business and Management (IOSRJBM) Volume 2, Issue 6 (Sep,-Oct. 2012), PP 10-13. International Journal of Engineering Science and Computing, July 2016 8268 http://ijesc.org/