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Minimal dominating functions of corona product graph of a cycle with a compl Minimal dominating functions of corona product graph of a cycle with a compl Document Transcript

  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 248 MINIMAL DOMINATING FUNCTIONS OF CORONA PRODUCT GRAPH OF A CYCLE WITH A COMPLETE GRAPH 1 M.Siva Parvathi, 2 B.Maheswari 1 Dept. of Mathematics, K.R.K. Govt. Degree College, Addanki-523201, Andhra Pradesh, India 2 Dept. of Applied Mathematics, S.P.Women’s University, Tirupat-517502, Andhra Pradesh, India ABSTRACT ‘Domination in graphs’ is the fastest growing area in Graph Theory that has emerged rapidly in the last three decades. An introduction and an extensive overview on domination in graphs and related topics is surveyed and detailed in the two books by Haynes et al. [ 1, 2]. Recently, dominating functions in domination theory have received much attention. Frucht and Harary [3] introduced a new product on two graphs G1 and G2, called corona product denoted by G1ÍG2. In this paper we present some results on minimal dominating functions of corona product graph ‫ܩ‬ ൌ ‫ܥ‬௡Í ‫ܭ‬௠. Key Words : Corona Product, Cycle, Complete graph, Dominating Function. Subject Classification: 1. INTRODUCTION Domination Theory is an important branch of Graph Theory that has many applications in Engineering, Communication Networks and many others. Allan, R.B. and Laskar, R.[4], Cockayne, E.J. and Hedetniemi, S.T. [5] have studied various domination parameters of graphs. An introduction and an extensive overview on domination in graphs and related topics is surveyed and detailed in the two books by Haynes et al. [ 1, 2]. Recently, dominating functions in domination theory have received much attention. A purely graph – theoretic motivation is given by the fact that the dominating function problem can be seen, in a clear sense, as a proper generalization of the classical domination problem. Jeelani Begum, S. [6] has studied some dominating functions of Quadratic Residue Cayley graphs. Product of graphs occur naturally in discrete mathematics as tools in combinatorial constructions. They give rise to important classes of graphs and deep structural problems. There are four main products that have been studied in the literature: the Cartesian product, the strong product, the direct product and the Lexicographic product of finite and infinite graphs. INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING & TECHNOLOGY (IJCET) ISSN 0976 – 6367(Print) ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), pp. 248-256 © IAEME: www.iaeme.com/ijcet.asp Journal Impact Factor (2013): 6.1302 (Calculated by GISI) www.jifactor.com IJCET © I A E M E
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 249 Frucht and Harary [3] introduced a new product on two graphs G1 and G2, called corona product denoted by G1ÍG2. The object is to construct a new and simple operation on two graphs G1 and G2 called their corona, with the property that the group of the new graph is in general isomorphic with the wreath product of the groups of G1 and of G2. In this paper we present some basic properties of corona product graph ‫ܩ‬ ൌ ‫ܥ‬௡Í ‫ܭ‬௠ and some results on minimal dominating functions are obtained. 2. CORONA PRODUCT OF ࡯࢔ AND ࡷ࢓ The corona product of a cycle ‫ܥ‬௡ with a complete graph ‫ܭ‬௠ is a graph obtained by taking one copy of a n – vertex graph ‫ܥ‬௡ and n copies of ‫ܭ‬௠ and then joining the ݅௧௛ vertex of ‫ܥ‬௡ to every vertex of ݅௧௛ copy of ‫ܭ‬௠ and it is denoted by ‫ܥ‬௡ ‫ܭ‬௠. We present some properties of the corona product graph ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ without proofs and the proofs can be found in Siva Parvathi, M. [7]. Theorem 2.1: The graph ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ is a connected graph. Theorem 2.2: The degree of a vertex v in G ൌ ‫ܥ‬௡ ‫ܭ‬௠ is given by ݀ሺ‫ݒ‬ሻ ൌ ൜ ݉ ൅ 2 , ݂݅ ‫ݒ‬ ‫א‬ ‫ܥ‬௡, ݉ , ݂݅ ‫ݒ‬ ‫א‬ ‫ܭ‬௠. Theorem 2.3: The number of vertices and edges in ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ is given respectively by 1. |ܸሺ‫ܩ‬ሻ| ൌ ݊ሺ݉ ൅ 1ሻ , 2. |‫ܧ‬ሺ‫ܩ‬ሻ| ൌ ௡ ଶ ሺ݉ଶ ൅ ݉ ൅ 2ሻ . Theorem 2.4: The graph ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ is non - hamiltonian. Theorem 2.5: The graph ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ is eulerian if ݉ is even. Theorem 2.6: The graph ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ is not bipartite. 3. DOMINATING SETS AND DOMINATING FUNCTIONS In this section we prove some results on Minimal Dominating Functions. First let us recall some definitions. Definition: Let ‫ܩ‬ ሺܸ, ‫ܧ‬ሻ be a graph. A subset ‫ܦ‬ of ܸ is said to be a dominating set (DS) of ‫ܩ‬ if every vertex in ܸ – ‫ܦ‬ is adjacent to some vertex in ‫.ܦ‬ A dominating set ‫ܦ‬ is called a minimal dominating set (MDS) if no proper subset of ‫ܦ‬ is a dominating set of ‫.ܩ‬ Definition: The domination number of ‫ܩ‬ is the minimum cardinality taken over all minimal dominating sets in ‫ܩ‬ and is denoted by γሺ‫ܩ‬ሻ. Definition: Let ‫ܩ‬ሺܸ, ‫ܧ‬ሻ be a graph. A function ݂: ܸ ՜ ሾ0,1ሿ is called a dominating function (DF) of ‫ܩ‬ if ݂ሺܰሾ‫ݒ‬ሿሻ ൌ ( ) [ ] ,1≥∑∈ vNu uf ݂‫ݎ݋‬ ݄݁ܽܿ ‫ݒ‬∈ܸ. Here ܰሾ‫ݒ‬ሿ is the closed neighbourhood set of ‫.ݒ‬ Definition: Let ݂ and ݃ be functions from ܸ to [0,1]. We define ݂ ൏ ݃ if ݂ሺ‫ݑ‬ሻ ≤ ݃ሺ‫ݑ‬ሻ for all ‫ݑ‬ ∈ ܸ, with strict inequality for atleast one vertex ‫.ݑ‬
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 250 A dominating function ݂ of ‫ܩ‬ is called a minimal dominating function (MDF) if for all ݃ ൏ ݂, ݃ is not a dominating function. Theorem 3.1: The domination number of ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ is n. i.e. ߛሺ‫ܩ‬ሻ ൌ ݊. Proof: Let ‫ܦ‬ denote a dominating set of ‫.ܩ‬ Case 1: Suppose ‫ܦ‬ contains the vertices of ‫ܥ‬௡ in ‫.ܩ‬ By the definition of ‫,ܩ‬ each vertex in ‫ܥ‬௡ is adjacent to all vertices of ith copy of ‫ܭ‬௠ . That is the vertices in ‫ܥ‬௡ dominate the vertices in all copies of ‫ܭ‬௠ respectively. Therefore the vertices of ‫ܦ‬ dominate all vertices of ‫.ܩ‬ Thus ‫ܦ‬ becomes a DS of ‫.ܩ‬ This set is also minimal, because, if we delete one vertex say ‫ݒ‬௞ from ‫,ܦ‬ then the vertices in the kth copy of ‫ܭ‬௠ are not dominated by any vertex in ‫.ܦ‬ Case 2: Suppose ‫ܦ‬ contains any vertex of each copy of ‫ܭ‬௠ in ‫.ܩ‬ That is |‫|ܦ‬ ൌ ݊. Obviously every vertex in ‫ܭ‬௠ dominates every other vertex in ‫ܭ‬௠ and also a single vertex of ‫ܥ‬௡ to which it is connected. Therefore the vertices in ‫ܦ‬ dominate all vertices of ‫.ܩ‬ Further this set is also minimal. Hence in either case we get ߛሺ‫ܩ‬ሻ ൌ ݊. Theorem 3.2: Let ‫ܦ‬ be a MDS of ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠. Then a function ݂ ‫׷‬ ܸ ՜ ሾ0, 1ሿ defined by ݂ሺ‫ݒ‬ሻ ൌ    ∈ otherwise.0, ,vif1, D becomes a MDF of ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠. Proof: Consider the graph ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ with vertex set V. Let ‫ܦ‬ be a MDS of G. Then by Theorem 3.1, ‫ܦ‬ contains either all the vertices of Cn or any vertex of each copy of Km in G. For definiteness let ‫ܦ‬ contain the vertices of ‫ܥ‬௡. Case 1: Let ‫ݒ‬ ‫א‬ ‫ܥ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in ‫.ܩ‬ Then ܰሾ‫ݒ‬ሿ contains m vertices of ‫ܭ‬௠ and three vertices of ‫ܥ‬௡ in ‫.ܩ‬ So [ ] ( ) .30.......0111 =+++++= −∈ ∑ 43421 timesmvNu uf Case 2: Let ‫ݒ‬ ‫א‬ ‫ܭ‬௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ in ‫.ܩ‬ Then ܰሾ‫ݒ‬ሿ contains m vertices of ‫ܭ‬௠ and one vertex of ‫ܥ‬௡ in ‫.ܩ‬ So [ ] ( ) 10.......01 =+++= −∈ ∑ 43421 timesmvNu uf . Therefore for all possibilities, we get [ ] ( ) ,1≥∑∈ uf vNu ∀ v ∈ V. This implies that ݂ is a DF. Now we check for the minimality of ݂. Define ݃ ‫׷‬ ܸ ՜ ሾ0, 1ሿ by ݃ሺ‫ݒ‬ሻ ൌ      ∈ ∈= otherwise.0, },{v-Dvif1, ,vvifr, k k D where 0 ൏ ‫ݎ‬ ൏ 1. Since strict inequality holds at the vertex ,Dvk ∈ it follows that ݃ ൏ ݂. Now the following cases arise. Case 1: Let ‫ݒ‬ ‫א‬ ‫ܥ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in ‫.ܩ‬ Sub case 1: Let ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ.
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 251 Then [ ] ( ) .120.......011 >+=+++++= −∈ ∑ rrug timesmvNu 43421 Sub case 2: Let ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ. Then [ ] ( ) .30.......0111 =+++++= −∈ ∑ 43421 timesmvNu ug Case 2: Let ‫ݒ‬ ‫א‬ ‫ܭ‬௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ in ‫.ܩ‬ Sub case 1: Let ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ. Then [ ] ( ) .10.......0 <=+++= −∈ ∑ rrug timesmvNu 43421 Sub case 2: Let ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ. Then [ ] ( ) .10.......01 =+++= −∈ ∑ 43421 timesmvNu ug This implies that [ ] ( ) 1<∑∈ ug vNu , for some ‫ݒ‬ ∈ ܸ. So ݃ is not a DF. Since ݃ is taken arbitrarily, it follows that there exists no ݃ ൏ ݂ such that ݃ is a DF. Thus ݂ is a MDF. Theorem 3.3: A function ݂ ‫׷‬ ܸ → [ 0, 1 ] defined by ݂ሺ‫ݒ‬ሻ ൌ ଵ ௤ାଵ , ∀ ‫ݒ‬ ∈ ܸ is a DF of ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠, if ‫ݍ‬ ൑ ݉. It is a MDF if ‫ݍ‬ ൌ ݉. Proof: Let V be the vertex set of ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ and ݂ be a function defined as in the hypothesis. Case 1: Suppose 0 ൏ ‫ݍ‬ ൏ ݉. Sub case 1: Let ‫ݒ‬ ‫א‬ ‫ܥ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in ‫.ܩ‬ Then ܰሾ‫ݒ‬ሿ contains ݉ vertices of ‫ܭ‬௠ and three vertices of ‫ܥ‬௡ in ‫.ܩ‬ So [ ] ( ) ( ) ,1 1 3 1 1 ....... 1 1 1 1 3 > + + = + ++ + + + = −+ ∈ ∑ q m qqq uf timesm vNu 4444 34444 21 since ‫ݍ‬ ൏ ݉. Sub case 2: Let ‫ݒ‬ ‫א‬ ‫ܭ‬௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ in ‫.ܩ‬ So [ ] ( ) ( ) ,1 1 1 1 1 ....... 1 1 1 1 1 > + + = + ++ + + + = −+ ∈ ∑ q m qqq uf timesm vNu 4444 34444 21 since ‫ݍ‬ ൏ ݉. Therefore for all possibilities, we get [ ] ( ) ,1>∑∈ uf vNu ∀ v ∈ V. This implies that ݂ is a DF. Now we check for the minimality of ݂. Define ݃ ‫׷‬ ܸ ՜ ሾ0, 1ሿ by ݃ሺ‫ݒ‬ሻ ൌ      + ∈= otherwise., 1q 1 ,vvifr, k V where 0 ൏ ‫ݎ‬ ൏ ଵ ௤ାଵ . Since strict inequality holds at a vertex ‫ݒ‬௞ of V, it follows that ݃ ൏ ݂. Case (i): Let ‫ݒ‬ ‫א‬ ‫ܥ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in G. Sub case 1: Let ‫ݒ‬௞ ‫א‬ Nሾvሿ.
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 252 Then [ ] ( ) ( ) 4444 34444 21 timesm vNu qqq rug −+ ∈ + ++ + + + +=∑ 2 1 1 ....... 1 1 1 1 1 1 3 1 2 1 1 > + + = + + + + < q m q m q , since ‫ݍ‬ ൏ ݉. Sub case 2: Let ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ. Then [ ] ( ) ( ) ,1 1 3 1 1 ....... 1 1 1 1 3 > + + = + ++ + + + = −+ ∈ ∑ q m qqq ug timesm vNu 4444 34444 21 since q ൏ ݉. Case (ii): Let ‫ݒ‬ ‫א‬ ‫ܭ‬௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ in G. Then we can see similarly as above that for the possibilities ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ, we have [ ] ( ) ,1 1 1 11 1 1 1 ....... 1 1 1 1 > + + =+ + + < + ++ + + + += − ∈ ∑ q m q m qqqq rug timesm vNu 4444 34444 21 since q ൏ ݉. and for ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ, we have [ ] ( ) ( ) ,1 1 1 1 1 ....... 1 1 1 1 1 > + + = + ++ + + + = −+ ∈ ∑ q m qqq ug timesm vNu 4444 34444 21 since q ൏ ݉. Hence it follows that [ ] ( ) ,1≥∑∈ ug vNu ∀ v ∈ V. Thus ݃ is a DF. This implies that ݂ is not a MDF. Case 2: Suppose ‫ݍ‬ ൌ ݉. Substituting q = m in Case 1 we can see that for ‫ݒ‬ ‫א‬ ‫ܥ‬௡ , we have [ ] ( ) ( ) .1 1 2 1 1 3 1 3 1 1 ....... 1 1 1 1 3 > + += + + = + + = + ++ + + + = −+ ∈ ∑ mm m q m qqq uf timesm vNu 4444 34444 21 and for ‫ݒ‬ ‫א‬ ‫ܭ‬௠, we have [ ] ( ) ( ) .1 1 1 1 1 1 1 ....... 1 1 1 1 1 = + + = + + = + ++ + + + = −+ ∈ ∑ m m q m qqq uf timesm vNu 4444 34444 21 Therefore for all possibilities, we get [ ] ( ) ,1≥∑∈ uf vNu ∀ v ∈ V. This implies that ݂ is a DF. Now we check for the minimality of ݂. Define ݃ ‫׷‬ ܸ ՜ ሾ0, 1ሿ by ݃ሺ‫ݒ‬ሻ ൌ     + ∈= otherwise., 1m 1 ,vvifr, k V where 0 ൏ ‫ݎ‬ ൏ ଵ ௠ାଵ . Since strict inequality holds at a vertex ‫ݒ‬௞ of V, it follows that ݃ ൏ ݂. Then we can see as in case (i), that for ‫ݒ‬ ‫א‬ ‫ܥ‬௡ and ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ , we have [ ] ( ) ( ) 44444 344444 21 timesm vNu mmm rug −+ ∈ + ++ + + + +=∑ 2 1 1 ....... 1 1 1 1 1 1 3 1 2 1 1 > + + = + + + + < m m m m m
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 253 and for ‫ݒ‬ ‫א‬ ‫ܥ‬௡ and ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ, we have [ ] ( ) ( ) .1 1 3 1 1 ....... 1 1 1 1 3 > + + = + ++ + + + = −+ ∈ ∑ m m mmm ug timesm vNu 44444 344444 21 Similarly we can show as in case (ii) that for ‫ݒ‬ ‫א‬ ‫ܭ‬௠ and ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ, we have [ ] ( ) .1 11 1 1 1 ....... 1 1 1 1 = + + + < + ++ + + + += − ∈ ∑ m m mmmm rug timesm vNu 44444 344444 21 and for ‫ݒ‬ ‫א‬ ‫ܭ‬௠ and ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ, we have [ ] ( ) ( ) .1 1 1 1 1 ....... 1 1 1 1 1 = + + = + ++ + + + = −+ ∈ ∑ m m mmm ug timesm vNu 44444 344444 21 This implies that [ ] ( ) 1<∑∈ ug vNu , for some v ∈ V. So ݃ is not a DF. Since ݃ is defined arbitrarily, it follows that there exists no ݃ ൏ ݂ such that ݃ is a DF. Thus ݂ is a MDF. Theorem 3.4: A function f : V → [ 0, 1 ] defined by ݂ሺ‫ݒ‬ሻ ൌ ௣ ௤ , ∀ ‫ݒ‬ ∈ ܸ where p = min (m, n) and q = max ( m, n ) is a DF if ௣ ௤ ൒ ଵ ௠ାଵ . Otherwise it is not a DF. Also it becomes MDF if ௣ ௤ ൌ ଵ ௠ାଵ . Proof: Let ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ be the given graph with vertex set ܸ. Let ݂ ‫׷‬ ܸ → ሾ 0, 1 ሿ be defined by ݂ሺ‫ݒ‬ሻ ൌ ௣ ௤ , ∀ ‫ݒ‬ ∈ܸ , where p = min(m, n) and q=max ( m, n). Clearly ௣ ௤ ൐ 0. Case 1: Let ‫ݒ‬ ‫א‬ ‫ܥ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in ‫ܩ‬. Then [ ] ( ) ( ) ( ) .3....... 3 q p m q p q p q p uf timesm vNu +=+++= −+ ∈ ∑ 444 3444 21 Case 2: Let ‫ݒ‬ ‫א‬ ‫ܭ‬௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ in G. Then [ ] ( ) ( ) ( ) .1....... 1 q p m q p q p q p uf timesm vNu +=+++= −+ ∈ ∑ 444 3444 21 From the above two cases, we observe that ݂ is a DF if ௣ ௤ ൒ ଵ ௠ାଵ . Otherwise ݂ is not a DF. Case 3: Suppose ௣ ௤ > ଵ ௠ାଵ . Clearly ݂ is a DF. Now we check for the minimality of ݂. Define ݃ ‫׷‬ ܸ ՜ ሾ0, 1ሿ by ݃ሺ‫ݒ‬ሻ ൌ      ∈= otherwise., q p ,vvifr, k V where 0 ൏ ‫ݎ‬ ൏ ௣ ௤ .
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 254 Since strict inequality holds at a vertex ‫ݒ‬௞ of V, it follows that ݃ ൏ ݂. Case (i): Let ‫ݒ‬ ‫א‬ ‫ܥ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in ‫ܩ‬. Sub case 1: Let ‫ݒ‬௞ ‫א‬ Nሾvሿ. Then [ ] ( ) ( ) 444 3444 21 timesm vNu q p q p q p rug −+ ∈ ++++=∑ 2 ....... ( ) ( ) ,132 >+=++< q p m q p m q p since ௣ ௤ > ଵ ௠ାଵ . Sub case 2: Let [ ]vNvk ∉ . Then [ ] ( ) ( ) ( ) .13....... 3 >+=+++= −+ ∈ ∑ q p m q p q p q p ug timesm vNu 444 3444 21 Case (ii): Let ‫ݒ‬ ‫א‬ ‫ܭ‬௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ in ‫ܩ‬. Sub case 1: Let ‫ݒ‬௞ ‫א‬ Nሾvሿ. Then [ ] ( ) 444 3444 21 timesm vNu q p q p q p rug − ∈ ++++=∑ ....... ( ) ,11 >+=+< q p m q p m q p since ௣ ௤ > ଵ ௠ାଵ . Sub case 2: Let [ ]vNvk ∉ . Then [ ] ( ) ( ) ( ) .11....... 1 >+=+++= −+ ∈ ∑ q p m q p q p q p ug timesm vNu 444 3444 21 Hence, it follows that [ ] ( ) ,1≥∑∈ ug vNu ∀ v ∈ V. Thus ݃ is a DF. This implies that ݂ is not a MDF. Case 4: Suppose ௣ ௤ ൌ ଵ ௠ାଵ . As in case 1 and 2, we have that [ ] ( ) ( ) ( ) ,1 1 2 1 1 1 33 > + += + +=+=∑∈ mm m q p muf vNu ݂݅ ‫ݒ‬ ‫א‬ ‫ܥ‬௡ . and [ ] ( ) ( ) ( ) ,1 1 1 11 = + +=+=∑∈ m m q p muf vNu ݂݅ ‫ݒ‬ ‫א‬ ‫ܭ‬௠. Therefore for all possibilities, we get [ ] ( ) ,1≥∑∈ uf vNu ∀ v ∈ V. This implies that ݂ is a DF. Now we check for the minimality of ݂. Define ݃ ‫׷‬ ܸ ՜ ሾ0, 1ሿ by ݃ሺ‫ݒ‬ሻ ൌ      ∈= otherwise., q p ,vvifr, k V where 0 ൏ ‫ݎ‬ ൏ ௣ ௤ . Since strict inequality holds at a vertex ‫ݒ‬௞ of V, it follows that ݃ ൏ ݂.
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 255 Then we can show as in case (i) that for ‫ݒ‬ ‫א‬ ‫ܥ‬௡ and ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ, we have [ ] ( ) ( ) .1....... 2 >++++= −+ ∈ ∑ 444 3444 21 timesm vNu q p q p q p rug and for ‫ݒ‬ ‫א‬ ‫ܥ‬௡ and ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ, we have [ ] ( ) ( ) .1....... 3 >+++= −+ ∈ ∑ 444 3444 21 timesm vNu q p q p q p ug Again we can show as in case (ii), that for ‫ݒ‬ ‫א‬ ‫ܭ‬௠ and ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ, we have [ ] ( ) .1....... <++++= − ∈ ∑ 444 3444 21 timesm vNu q p q p q p rug and for ‫ݒ‬ ‫א‬ ‫ܭ‬௠ and ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ, we have [ ] ( ) ( ) ( ) ( ) .1 1 1 11....... 1 = + +=+=+++= −+ ∈ ∑ m m q p m q p q p q p ug timesm vNu 444 3444 21 This implies that [ ] ( ) 1<∑∈ ug vNu , for some v ∈ V. So ݃ is not a DF. Since ݃ is defined arbitrarily, it follows that there exists no ݃ ൏ ݂ such that ݃ is a DF. Thus ݂ is a MDF. ILLUSTRATION 1 1 11 1 0 0 0 0 0 0 0 0 00 00 0 0 0 0 0 0 00 ࡯૞ ٖ ࡷ૝ The function f takes the value 1 for vertices of Cn and value 0 for vertices of Km in each copy.
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 256 REFERENCES 1. Haynes, T.W., Hedetniemi, S.T. and Slater, P.J. (1998), Domination in Graphs: Advanced Topics, Marcel Dekker, Inc., New York. 2. Haynes, T.W., Hedetniemi, S.T. and Slater, P.J. (1998), Fundamentals of domination in graphs, Marcel Dekker, Inc., New York . 3. Frucht, R. and Harary, F.(1970), On the corona of Two Graphs, Aequationes Mathematicae, Volume 4, Issue 3, pp. 322 – 325. 4. Allan, R.B. and Laskar, R.C. (1978), On domination, independent domination numbers of a graph. Discrete Math., 23, 73 – 76. 5. Cockayne, E.J. and Hedetniemi, S.T. (1977), Towards a theory of domination in graphs. Networks, 7, 247 – 261. 6. Jeelani Begum, S. (2011), Some studies on dominating functions of Quadratic Residue Cayley Graphs, Ph. D. thesis, Sri Padmavathi Mahila Visvavidyalayam, Tirupati, Andhra Pradesh, India. 7. Siva Parvathi, M. (2013), Some studies on dominating functions of Corona Product Graphs, Ph. D. thesis to be submitted to Sri Padmavathi Mahila Visvavidyalayam, Tirupati, Andhra Pradesh, India. 8. László Lengyel, “The Role of Graph Transformations in Validating Domain-Specific Properties”, International Journal of Computer Engineering & Technology (IJCET), Volume 3, Issue 3, 2012, pp. 406 - 425, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375. 9. Syed Abdul Sattar, Mohamed Mubarak.T, Vidya Pv and Appa Rao, “Corona Based Energy Efficient Clustering in WSN”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 3, 2013, pp. 233 - 242, ISSN Print: 0976-6480, ISSN Online: 0976-6499.