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• International Journal of Information TechnologyINFORMATION TECHNOLOGY & INTERNATIONAL JOURNAL OF & Management Information System (IJITMIS), ISSN 0976 – 6405(Print), ISSN 0976 – 6413(Online), Volume 5, Issue 1, January - February (2014), © IAEME MANAGEMENT INFORMATION SYSTEM (IJITMIS) ISSN 0976 – 6405(Print) ISSN 0976 – 6413(Online) Volume 5, Issue 1, January - February (2014), pp. 01-11 © IAEME: http://www.iaeme.com/IJITMIS.asp Journal Impact Factor (2013): 5.2372 (Calculated by GISI) www.jifactor.com IJITMIS ©IAEME MINIMAL DOMINATING FUNCTIONS OF CORONA PRODUCT GRAPH OF A PATH WITH A STAR 1 1 Makke Siva Parvathi, 2 Bommireddy Maheswari 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 an emerging area of research in recent years and has been studied extensively. 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 is receiving much attention. They give rise to an important classes of graphs and deep structural problems. In this paper we discuss some results on minimal dominating functions of corona product graph of a path with a star. Key Words: Corona Product, Path, Star, Dominating Function. Subject Classification: 68R10 1. INTRODUCTION Domination Theory is an important branch of Graph Theory that has wide range of applications to ma ny fiel ds li ke Engineering, Communication Networks, S o c i a l S c i e n c e s , l i n g u i s t i c s , P h y s i c a l S c i e n c e s and many others. V arious domination parameters of graphs are studied by Allan, R.B. and Laskar, R.[3]. The concept of minimal dominating functions was introduced by Cockayne, E.J. and Hedetniemi, S.T. [4] and Cockayne et al. [5]. Jeelani Begum, S. [6] has studied some dominating functions of Quadratic Residue Cayley graphs. Frucht and Harary [7] introduced a new product on two graphs G1 and G2 called corona product denoted by G1 Í G2. 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 1
• International Journal of Information Technology & Management Information System (IJITMIS), ISSN 0976 – 6405(Print), ISSN 0976 – 6413(Online), Volume 5, Issue 1, January - February (2014), © IAEME the new graph is in general isomorphic with the wreath product of the groups of G1 and of G2 . The authors have studied some dominating functions of corona product graph of a cycle with a complete graph [8] and published papers on minimal dominating functions, some variations of Y – dominating functions and Y – total dominating functions [9,10,11]. In this paper we present some basic properties of corona product graph of a path with a star and some results on minimal dominating functions. 2. CORONA PRODUCT OF ࡼ࢔ AND ࡷ૚,࢓ The corona product of a path ܲ with star ‫ܭ‬ଵ,௠ is a graph obtained by taking one ௡ copy of a ݊ – vertex path ܲ and n copies of ‫ܭ‬ଵ,௠ and then joining the ݅௧௛ vertex of ܲ to ௡ ௡ every vertex of ݅ ௧௛ copy of ‫ܭ‬ଵ,௠ and it is denoted by ܲ ‫ܭ‬ଵ,௠ . ௡ We present some properties of the corona product graph ܲ ‫ܭ‬ଵ,௠ without proofs ௡ and the proofs can be found in Siva Parvathi, M. [8]. Theorem.2.1 : The graph ‫ ܩ‬ൌ ܲ ‫ܭ‬ଵ,௠ is a connected graph. ௡ Theorem 2.2: The degree of a vertex ‫ݒ‬௜ in ‫ ܩ‬ൌ ܲ௡ ‫ܭ‬ଵ,௠ is given by ݉ ൅ 3, ݂݅ ‫ݒ‬௜ ‫ܲ א‬௡ ܽ݊݀ 2 ൑ ݅ ൑ ሺ݊ െ 1ሻ, ‫ۓ‬ ݉ ൅ 2, ݂݅ ‫ݒ‬௜ ‫ܲ א‬௡ ܽ݊݀ ݅ ൌ 1 ‫,݊ ݎ݋‬ ݀ሺ‫ݒ‬௜ ሻ ൌ ݂݅ ‫ݒ‬௜ ‫ܭ א‬ଵ,௠ ܽ݊݀ ‫ݒ‬௜ ݅‫,݊݋݅ݐ݅ݐݎܽ݌ ݐݏݎ݂݅ ݊݅ ݏ‬ ‫ ݉ ۔‬൅ 1, 2, ݂݅ ‫ݒ‬௜ ‫ܭ א‬ଵ,௠ ܽ݊݀ ‫ݒ‬௜ ݅‫.݊݋݅ݐ݅ݐݎܽ݌ ݀݊݋ܿ݁ݏ ݊݅ ݏ‬ ‫ە‬ Theorem 2.3: The number of vertices and edges in ‫ ܩ‬ൌ ܲ ‫ܭ‬ଵ,௠ is given by ௡ 1. |ܸሺ‫ܩ‬ሻ| ൌ ݊ ሺ݉ ൅ 2ሻ, 2. |‫ܧ‬ሺ‫ܩ‬ሻ| ൌ 2݊ ሺ݉ ൅ 1ሻ – 1. Theorem 2.4: The graph ‫ ܩ‬ൌ ܲ௡ ‫ܭ‬ଵ,௠ is not eulerian. Theorem 2.5: The graph ‫ ܩ‬ൌ ܲ௡ ‫ܭ‬ଵ,௠ is non hamiltonian. Theorem 2.6: The graph ‫ ܩ‬ൌ ܲ௡ ‫ܭ‬ଵ,௠ is not bipartite. 3. DOMINATING SETS AND DOMINATING FUNCTIONS In this section we discuss dominating sets (DS), dominating functions (DFs) of the graph ‫ ܩ‬ൌ ܲ௡ ‫ܭ‬ଵ,௠ . We present some results related to minimal dominating functions of this graph. Theorem 3.1: The domination number of ‫ ܩ‬ൌ ܲ௡ ‫ܭ‬ଵ,௠ is ݊. Proof: Let D denote a dominating set of G. Case 1: Suppose D contains the vertices of path ܲ in G. ௡ By the definition of the graph G which is the corona product of ܲ and ‫ܭ‬ଵ,௠ , the ݅ ௧௛ ௡ vertex ‫ݒ‬௜ in ܲ௡ is adjacent to all vertices of ith copy of ‫ܭ‬ଵ,௠ . That is the vertices in ܲ ௡ dominate the vertices in all copies of ‫ܭ‬ଵ,௠ respectively. Therefore the vertices in D dominate all vertices of G. Thus D becomes a DS of G. Case 2: Suppose D contains a vertex of degree ݉ ൅ 1 in each copy of ‫ܭ‬ଵ,௠ in G. As there are n copies of ‫ܭ‬ଵ,௠ , it follows that |‫ |ܦ‬ൌ ݊. Obviously these vertices in G dominate the vertices in all copies of ‫ܭ‬ଵ,௠ respectively and also a single vertex of ܲ௡ to which it is connected. Therefore the vertices of D dominate all vertices of G. Further this set is also minimal. Thus ߛሺ‫ܩ‬ሻ ൌ ݊. 2
• International Journal of Information Technology & Management Information System (IJITMIS), ISSN 0976 – 6405(Print), ISSN 0976 – 6413(Online), Volume 5, Issue 1, January - February (2014), © IAEME Theorem 3.2: Let D be a MDS of ‫ ܩ‬ൌ ܲ௡ ݂ ሺ‫ ݒ‬ሻ ൌ  ‫ܭ‬ଵ,௠ . Then a function ݂: ܸ ՜ ሾ0,1ሿ defined by 1, if v ∈ D, 0, otherwise. becomes a MDF of ‫ ܩ‬ൌ ܲ ‫ܭ‬ଵ,௠ . ௡ Proof: Consider the graph ‫ ܩ‬ൌ ܲ௡ ‫ܭ‬ଵ,௠ with vertex set V. We have seen in Theorem 3.1 that a DS of ‫ ܩ‬contains all the vertices of ܲ and this ௡ set is minimal. Also the set of vertices whose degree is ሺ݉ ൅ 1ሻ in each copy of ‫ܭ‬ଵ,௠ form a minimal DS of G. Let D be a MDS of ‫ .ܩ‬For definiteness, let D contain the vertices of ܲ௡ in ‫ .ܩ‬In ܲ , there are ௡ two end vertices whose degree is ݉ ൅ 2 and there are ݊ – 2 intermediate vertices whose degree is ݉ ൅ 3 respectively in ‫ .ܩ‬In ‫ܭ‬ଵ,௠ , there is one vertex of degree ݉ ൅ 1 and there are ݉ vertices whose degree is 2 respectively in ‫.ܩ‬ Case 1: Let ‫ ܲ א ݒ‬be such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 3 in G. ௡ Then   ∑[ ] f (u ) = 1 + 1 + 1 + 0 +4243  = 3. 1 ....... + 0 Case 2: Let ‫ ܲ א ݒ‬be such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 2 in G. ௡   u∈ N v Then   ( m +1) − times   ∑[ ] f (u ) = 1 + 1 + 0 +4243  = 2. 1 ....... + 0 Case 3: Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 1 in G.   u∈N v Then ( m +1) − times     ∑[ ] f (u ) = 1 + 0 +4243  = 1 . 1 ....... + 0 Case 4: Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ 2 in G. Then ∑ f (u ) = 1 + 0 + 0 = 1. u∈ N v   ( m +1) − times   u∈N [v ] ∑ f (u ) ≥ 1, ∀ Therefore for all possibilities, we get This implies that ݂ is a DF. Now we check for the minimality of ݂. Define g : V → [0,1] by v ∈ V. u∈N [v ] r, for v = v k ∈ D, ݃ሺ‫ݒ‬ሻ ൌ 1, for v ∈ D - {v k },  0, otherwise.  where 0 ൏ ‫ ݎ‬൏ 1. Since strict inequality holds at the vertex vk ∈ D, it follows that ݃ ൏ ݂. Case (i): Let ‫ ܲ א ݒ‬be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 3 in G. ௡ Sub case 1: Let vk ∈ N [v] . Then   ∑[ ] g (u ) = r + 1 + 1 + 0 +4243  = r + 2 > 1. 1 ....... + 0 u∈N v   ( m +1) − times   Sub case 2: Let vk ∉ N [v] . 3
• International Journal of Information Technology & Management Information System (IJITMIS), ISSN 0976 – 6405(Print), ISSN 0976 – 6413(Online), Volume 5, Issue 1, January - February (2014), © IAEME Then   ∑[ ] g (u ) = 1 + 1 + 1 + 0 +4243  = 3. 1 ....... + 0 Case (ii): Let ‫ܲ א ݒ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in G. Sub case 1: Let vk ∈ N [v] .   u∈N v Then   ( m +1) − times   ∑[ ] g (u ) = r + 1 + 0 +4243  = r + 1 > 1. 1 ....... + 0   u∈N v   ( m +1) − times Sub case 2: Let vk ∉ N [v] . Then   ∑[ ] g (u ) = 1 + 1 + 0 +4243  = 2. 1 ....... + 0 Case (iii): Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 1 in G. Sub case 1: Let vk ∈ N [v] .   u∈N v Then ( m +1) − times     ∑[ ] g (u ) = r + 0 +4243  = r < 1. 1 ....... + 0   u∈N v ( m +1) − times   Sub case 2: Let vk ∉ N [v] . Then   ∑[ ] g (u ) = 1 + 0 +4243  = 1. 1 ....... + 0 Case (iv): Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ ሺ‫ ݒ‬ሻ ൌ 2 in G. Sub case 1: Let vk ∈ N [v] . Then ∑ g (u ) = r + 0 + 0 = r < 1.   u∈ N v ( m +1) − times   u∈N [v ] Sub case 2: Let vk ∉ N [v] . Then ∑ g (u ) = 1 + 0 + 0 = 1. u∈N [v ] This implies that ∑ g (u ) < 1, for some v ∈V. So ݃ is not a DF. Since ݃ is taken arbitrarily, it follows that there exists no ݃ ൏ ݂ such that ݃ is a DF. Thus ݂ is a MDF. ଵ Theorem 3.3: A function ݂ ‫ → ܸ ׷‬ሾ 0, 1 ሿ defined by ݂ ሺ‫ ݒ‬ሻ ൌ ௤ , ∀ ‫ ܸ ∈ ݒ‬is a DF of u∈N [v ] ‫ ܩ‬ൌ ܲ ‫ܭ‬ଵ,௠ , if ‫ ݍ‬൑ 3. It is a MDF if ‫ ݍ‬ൌ 3. ௡ Proof: Let ݂ be a function defined as in the hypothesis. Case I: Suppose ‫ ݍ‬൏ 3. Case 1: Let ‫ ܲ א ݒ‬be such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 3 in G. ௡ So 1 1 1 ∑[ ] f (u ) = q + q + ....... + q = u∈N v 1442443 m+4 > 1, since ‫ ݍ‬൏ 3 and ݉ ൒ 2. q Case 2: Let ‫ ܲ א ݒ‬be such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 2 in G. ௡ ( m + 4 )−times So 1 1 1 ∑[ ] f (u ) = q + q + ....... + q = u∈N v 1442443 m+3 > 1, q since ‫ ݍ‬൏ 3 and ݉ ൒ 2. Case 3: Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 1 in G. ( m + 3 )−times 4
• International Journal of Information Technology & Management Information System (IJITMIS), ISSN 0976 – 6405(Print), ISSN 0976 – 6413(Online), Volume 5, Issue 1, January - February (2014), © IAEME 1 So 1 1 ∑[ ] f (u ) = q + q + ....... + q = u∈N v 1442443 m+2 > 1, since ‫ ݍ‬൏ 3 and ݉ ൒ 2. q Case 4: Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ 2 in G. 1 1 1 3 So ∑ f (u ) = + + = > 1, since ‫ ݍ‬൏ 3. q q q q u∈N [v ] Therefore for all possibilities, we get ∑ f (u ) > 1, ∀ v ∈ V. ( m + 2 )−times This implies that ݂ is a DF. Now we check for the minimality of ݂. Define g : V → [0,1] by u∈N [v ]  r, if v = v k ∈ V ,  ݃ሺ‫ ݒ‬ሻ ൌ  1  q , otherwise.  where 0 ൏ ‫ ݎ‬൏ ௤ . ଵ Since strict inequality holds at a vertex ‫ݒ‬௞ of V, it follows that ݃ ൏ ݂. Case (i): Let ‫ ܲ א ݒ‬be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 3 in G. ௡ Sub case 1: Let vk ∈ N[v]. 1 1 1 Then ∑ g (u ) = r + + + ....... + < 1 + m + 3 = m + 4 > 1, since ‫ ݍ‬൏ 3 and ݉ ൒ 2. q q q q q q u∈N [v ] 1442443 ( m+3)−times Sub case 2: Let vk ∉ N [v] . The 1 1 1 ∑[ ] g (u ) = q + q + ....... + q = u∈N v 1442443 m+4 > 1, since ‫ ݍ‬൏ 3 and ݉ ൒ 2. q Case (ii): Let ‫ܲ א ݒ‬௡ be such that ݀ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 2 in G. Sub case 1:Let vk ∈ N [v] . 1 m+2 m+3 Then ∑ g (u ) = r + 1 + 1 + ....... + 1 < + = > 1, since ‫ ݍ‬൏ 3 and ݉ ൒ 2. q q q q q q u∈N [v ] ( m + 4 )−times 1442443 (m + 2 )−times Sub case 2: Let vk ∉ N [v] . Then 1 1 1 ∑[ ] g (u ) = q + q + ....... + q = u∈N v 1442443 m+3 > 1, since ‫ ݍ‬൏ 3 and ݉ ൒ 2. q Case (iii): Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 1 in G. Sub case 1:Let vk ∈ N [v] . ( m + 3 )−times Then 1 1 1 ∑[ ] g (u ) = r + q + q + ....... + q u∈N v 1442443 ݉ ൒ 2. Sub case 2:Let vk ∉ N [v] . < 1 m +1 m + 2 + = ≥ 1, q q q (m +1)−times 5 since ‫ ݍ‬൏ 3 and
• International Journal of Information Technology & Management Information System (IJITMIS), ISSN 0976 – 6405(Print), ISSN 0976 – 6413(Online), Volume 5, Issue 1, January - February (2014), © IAEME 1 So 1 1 ∑[ ] g (u ) = q + q + ....... + q = u∈N v 1442443 m+2 > 1, since ‫ ݍ‬൏ 3 and ݉ ൒ 2. q Case (iv): Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ ሺ‫ ݒ‬ሻ ൌ 2 in G. Sub case 1: Let vk ∈ N [v]. ( m + 2 )−times Then 1 1 3 ∑[ ] g (u ) = r + q + q < q , since ‫ ݍ‬൏ 3, it follows that u∈N v 3 > 1. q Sub case 2: Let vk ∉ N [v] . 1 1 1 3 Then ∑ g (u ) = + + = > 1, since ‫ ݍ‬൏ 3. q q q q u∈N [v ] ∑ g (u ) ≥ 1, ∀v ∈V . Hence, it follows that Thus ݃ is a DF. This implies that ݂ is not a MDF. Case II: Suppose ‫ ݍ‬ൌ 3. Substituting q = 3 in case I, for ‫ ܲ א ݒ‬such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 3, we have ௡ u∈N [v ] 1 1 1 ∑[ ] f (u ) = q + q + ....... + q = u∈N v 1442443 m+4 m+4 m +1 = = 1+ > 1, 3 3 q and for ‫ ܲ א ݒ‬such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2, we have ௡ ( m + 4 )−times 1 1 1 ∑[ ] f (u ) = q + q + ....... + q = u∈N v 1442443 m+3 m+3 m = = 1 + > 1. q 3 3 Also for ‫ܭ א ݒ‬ଵ,௠ such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 1, we have 1 1 1 m+2 m+2 ∑[v] f (u ) = q + q + ....... + q = q = 3 > 1, since ݉ ൒ 2. u∈N 1442443 ( m +3 )−times And for ‫ܭ א ݒ‬ଵ,௠ such that ݀ ሺ‫ ݒ‬ሻ ൌ 2, we have (m + 2 )−times 1 1 1 3 3 ∑[ ] f (u ) = q + q + q = q = 3 = 1. u∈N v Therefore for all possibilities, we get ∑ f (u ) ≥ 1, ∀v ∈V . This implies that ݂ is a DF. Now we check for the minimality of ݂. Define g : V → [0,1] by u∈N [v ]  r, for v = v k ∈ V ,  ݃ ሺ‫ ݒ‬ሻ ൌ  1  q , otherwise.  ଵ where 0 ൏ ‫ ݎ‬൏ ௤ . Since strict inequality holds at a vertex ‫ݒ‬௞ of V, it follows that ݃ ൏ ݂. Then we can show as in case (i) that for ‫ ܲ א ݒ‬such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 3, ௡ 6
• International Journal of Information Technology & Management Information System (IJITMIS), ISSN 0976 – 6405(Print), ISSN 0976 – 6413(Online), Volume 5, Issue 1, January - February (2014), © IAEME 1 1 1 ∑[ ] g (u ) = r + q + q + ....... + q > 1, u∈N v if vk ∈ N [v], 1442443 ( m+ 3 )−times and if vk ∉ N [v] . 1 1 1 m+4 m+4 m +1 ∑[v ] g (u ) = q + q + ....... + q = q = 3 = 1 + 3 > 1, u∈N 1442443 Again as in case (ii), for ‫ܲ א ݒ‬௡ such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 2, (m + 4 )−times We have 1 1 1 ∑[ ] g (u ) = r + q + q + ....... + q > 1, u∈N v if vk ∈ N [v] . 1442443 ( m+ 2 )−times And ∑ g (u ) = 1 + 1 + ....... + 1 = m + 3 = m + 3 = 1 + m > 1, if vk ∉ N [v] . 3 3 q q q q u∈N [v ] 1442443 Again we can see as in case (iii) that for ‫ܭ א ݒ‬ଵ,௠ such that ݀ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 1, ( m+3 )−times 1 1 1 ∑ ] g (u ) = r + q + q + .......+ q > 1, [ u∈N v if vk ∈ N [v] . 144 44 2 3 (m +1)−times 1 1 1 m+2 m+2 ∑[v ] g (u ) = q + q + ....... + q = q = 3 > 1, if vk ∉ N [v]. u∈N 1442443 And (m + 2 )−times Similarly we can show as in case (iv) that for 1 1 3 ∑[v ] g (u ) = r + q + q < q = 3 = 1, if vk ∈ N [v], 3 u∈N and 1 1 1 3 3 ‫ܭ א ݒ‬ଵ,௠ such that ݀ሺ‫ݒ‬ሻ ൌ 2, vk ∉ N [v]. ∑[ ] g (u ) = q + q + q = q = 3 = 1, if u∈N v ∑ g (u ) < 1 , for some This implies that 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 ݂ ‫ ]1 ,0[ → ܸ ׷‬defined by ݂ሺ‫ݒ‬ሻ ൌ , ∀ ‫ ܸ ∈ ݒ‬where u∈N [v ] ‫ ݌‬ൌ ݉݅݊ ሺ݉, ݊ሻ and ‫ ݍ‬ൌ ݉ܽ‫ ݔ‬ሺ݉, ݊ሻ is a DF of ‫ ܩ‬ൌ ܲ ௡ Otherwise it is not a DF. Also it becomes MDF if ௣ ௤ ௣ ൌ . ଵ ଷ ௤ ‫ܭ‬ଵ,௠ , if ௣ ௤ ൒ ଵ ଷ . Proof: Let ݂: ܸ → [ 0, 1 ] be defined by ݂ሺ‫ݒ‬ሻ ൌ , ∀ ‫ ܸ ∈ ݒ‬where ‫ ݌‬ൌ ݉݅݊ሺ݉, ݊ሻ and ‫ ݍ‬ൌ ݉ܽ‫ ݔ‬ሺ݉, ݊ሻ. Clearly ௣ ௤ ௤ ൐ 0. Case 1: Let ‫ ܲ א ݒ‬be such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 3 in ‫.ܩ‬ ௡ p p p p So f (u ) = + + ....... + = (m + 4 ) . ∑ u∈N [v ] q q q 144 244 3 4 4 q Case 2: Let ‫ ܲ א ݒ‬be such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 2 in ‫.ܩ‬ ௡ (m + 4 )−times 7
• International Journal of Information Technology & Management Information System (IJITMIS), ISSN 0976 – 6405(Print), ISSN 0976 – 6413(Online), Volume 5, Issue 1, January - February (2014), © IAEME So p p p p ∑[ ] f (u ) = q + q + ....... + q = (m + 3) q . u∈N v 1442443 4 4 Case 3: Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 1 in ‫. ܩ‬ p p p p So f (u ) = + + ....... + = (m + 2 ) . (m+3)−times ∑ q q q 144244 3 4 4 u∈N [v ] q Case 4: Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ 2 in ‫.ܩ‬ p p p p So ∑ f (u ) = + + = 3 . q q q q u∈N [v ] (m + 2 )−times From the above four cases, we observe that ݂ is a DF if Otherwise ݂ is not a DF. ௣ Case 5: Suppose ൐ ௤ ଵ ଷ ൒ . ௣ ଵ ௤ ଷ Clearly ݂ is a DF. Now we check for the minimality of ݂. Define g : V → [0,1] by .  r, for v = v k ∈ V ,  ݃ሺ‫ ݒ‬ሻ ൌ  p  q , otherwise.  ௣ where 0 ൏ ‫ ݎ‬൏ . ௤ Since strict inequality holds at a vertex ‫ݒ‬௞ of V, it follows that ݃ ൏ ݂ . Case (i): Let ‫ ܲ א ݒ‬be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 3 in ‫. ܩ‬ ௡ Sub case 1: Let vk ∈ N [v]. p p p p p p Then ∑ g (u ) = r + + + ....... + < + (m + 3) = (m + 4 ) > 1, since q q q q q q u∈N [v ] 1442443 4 4 ௣ ௤ ൐ ଵ ଷ and m ≥ ( m+3)−times 2. Sub case 2: Let v k ∉ N [v ] . p p p p Then ∑ g (u ) = + + ....... + = (m + 4) > 1, since q q q q u∈N [v ] 1442443 4 4 ௣ ௤ ൐ ଵ ଷ and m ≥ 2. Case (ii): Let ‫ܲ א ݒ‬௡ be such that ݀ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 2 in ‫.ܩ‬ Sub case 1: Let vk ∈ N [v] . (m + 4 )−times Then p p p p p p ∑[ ] g (u ) = r + q + q + ....... + q < q + (m + 2) q = (m + 3) q > 1, u∈N v since 1442443 4 4 ( m+ 2 )−times 2. Sub case 2: Let vk ∉ N [v] . Then p p p p ∑[ ] g (u ) = q + q + ....... + q = (m + 3) q > 1, since u∈N v 1442443 4 4 ௣ ௤ Case (iii): Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 1 in ‫. ܩ‬ ( m +3 )−times 8 ൐ ଵ ଷ and m ≥ 2. ௣ ௤ ൐ ଵ ଷ and m ≥
• International Journal of Information Technology & Management Information System (IJITMIS), ISSN 0976 – 6405(Print), ISSN 0976 – 6413(Online), Volume 5, Issue 1, January - February (2014), © IAEME Sub case 1: Let vk ∈ N [v] . p p p p p p Then ∑ g (u ) = r + + + ....... + < + (m + 1) = (m + 2) > 1, since q q q q q u∈N [v ] 1442443 q 4 4 ௣ ௤ ( m+1)−times Sub case 2: Let vk ∉ N [v]. p p p p Then ∑ g (u ) = + + ....... + = (m + 2) > 1, since q q q q u∈N [v ] 1442443 4 4 ௣ ௤ ൐ ଵ ଷ and m ≥ 2. Case (iv): Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ ሺ‫ ݒ‬ሻ ൌ 2 in ‫.ܩ‬ Sub case 1: Let vk ∈ N [v]. ( m + 2 )−times Then p p p > 1, since q ∑[ ] g (u ) = r + q + q < 3 u∈N v ௣ ௤ Sub case 2: Let vk ∉ N [v] . Then p p p ∑[ ] g (u ) = q + q + q u∈ N v ∑ g (u ) > 1, ∀ Hence, it follows that ௣ p > 1, since q =3 ൐ v ∈ V. ௤ ଵ ଷ . ൐ ଵ ଷ . Thus ݃ is a DF. This implies that ݂ is not a MDF. ௣ ଵ Case 6: Suppose ൌ . u∈N [v ] ௤ ଷ As in case 1, we can show for ‫ ܲ א ݒ‬such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 3, ௡ m +1  p 1 p p p ∑[v ] f (u ) = q + q + ....... + q = (m + 4) q  = (m + 4) 3  = 1 + 3 > 1.     u∈N   1442443 4 4 Again as in case 2, for ‫ܲ א ݒ‬௡ such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 2, we have  p p p p m 1 f (u ) = + + ....... + = (m + 3)  = (m + 3)  = 1 + > 1 . (m + 4 )−times ∑ u∈N [v ] q q q 1442443 4 4 q   3 3 We can show as in case 3, for ‫ܭ א ݒ‬ଵ,௠ such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 1, .  p p p p m −1 1 >1 ∑ f (u ) = + + ....... + = (m + 2)  = (m + 2)  = 1 +   (m + 3 )−times u∈N [v ] q q q 1442443 4 4 3 q 3 Again as in case 4, we can show for ‫ܭ א ݒ‬ଵ,௠ such that ݀ሺ‫ݒ‬ሻ ൌ 2, p p p p 1 ∑[v ] f (u ) = q + q + q = 3 q = 3 3 = 1. u∈N Therefore for all possibilities, we get ∑ f (u ) ≥ 1, ∀ v ∈ V. (m + 2 )−times This implies that ݂ is a DF. Now we check for the minimality of ݂ . Define g : V → [0,1] by u∈N [v ]  r, for v = v k ∈ V ,  ݃ሺ‫ݒ‬ሻ ൌ  p  q , otherwise.  9 ൐ ଵ ଷ and m ≥ 2.
• International Journal of Information Technology & Management Information System (IJITMIS), ISSN 0976 – 6405(Print), ISSN 0976 – 6413(Online), Volume 5, Issue 1, January - February (2014), © IAEME where 0 ൏ ‫ ݎ‬൏ . ௣ ௤ Since strict inequality holds at a vertex ‫ݒ‬௞ of V, it follows that ݃ ൏ ݂. Then we can show as in case (i), (ii) and (iii) of case 5 that ∑[ ] g (u ) > 1, if ‫ܲ א ݒ‬ ௡ and vk ∈ N [v] or vk ∉ N [v] . u∈N v Now we can show as in case (iv) of case 5 that ∑[ ] g (u ) = r + q + q < 1, if ‫ܭ א ݒ‬ଵ,௠ and p u∈N v And p vk ∈ N [v] . ∑[ ] g (u ) = q + q + q = 3 q  = 3 3  = 1, if ‫ܭ א ݒ‬ଵ,௠ and     p p  p  u∈N v This implies that p 1  ∑ g (u ) < 1, for some v∈V. So ݃ is not a DF. vk ∉ N [v] . Since ݃ is defined arbitrarily, it follows that there exists no ݃ ൏ ݂ such that ݃ is a DF. Thus ݂ is a MDF. u∈N [v ] ILLUSTRATION 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 ࡼ૞ ٖ ࡷ૚,૜ 0 0 The function f takes the value 1 for vertices of P5 and value 0 for vertices in each copy of K1,3 10
• International Journal of Information Technology & Management Information System (IJITMIS), ISSN 0976 – 6405(Print), ISSN 0976 – 6413(Online), Volume 5, Issue 1, January - February (2014), © IAEME REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] Haynes, T.W., Hedetniemi, S.T. and Slater, P.J. - Domination in Graphs: Advanced Topics, Marcel Dekker, Inc., New York, (1998). Haynes, T.W., Hedetniemi, S.T. and Slater, P.J. - Fundamentals of domination in graphs, Marcel Dekker, Inc., New York , (1998). Allan, R.B. and Laskar, R.C. – On domination, independent domination numbers of a graph. Discrete Math., 23 (1978), 73 – 76. Cockayne, E.J. and Hedetniemi, S.T. - Towards a theory of domination in graphs. Networks, 7 (1977), 247 – 261. Cockayne, E.J., Mynhardt, C.M. and Yu, B. - Total dominating functions in trees: Minimality and Convexity, Journal of Graph Theory, 19(1995), 83 – 92. Jeelani Begum, S. - Some studies on dominating functions of Quadratic Residue Cayley Graphs, Ph. D. thesis, Sri Padmavathi Mahila Visvavidyalayam, Tirupati, Andhra Pradesh, India, (2011). Frucht, R. and Harary, F. - On the corona of Two Graphs, Aequationes Mathematicae, Volume 4, Issue 3 (1970), 322 – 325. Siva Parvathi, M – Some studies on dominating functions of corona product graphs, Ph.D thesis, Sri Padmavati Mahila Visvavidyalayam, Tirupati, Andhra Pradesh, India, (2013). Siva Parvathi, M and Maheswari, B. - Minimal Dominating Functions of Corona Product Graph of a Cycle with a Complete Graph - International Journal of Computer Engineering & Technology, Volume 4, Issue 4 (2013), 248 – 256. Siva Parvathi, M and Maheswari, B. - Some variations of Y-Dominating Functions of Corona Product Graph of a Cycle with a Complete Graph- International Journal of Computer Applications, Volume 81, Issue 1 (2013), 16 – 21. Siva Parvathi, M and Maheswari, B. - Some variations of Total Y-Dominating Functions of Corona Product Graph of a Cycle with a Complete Graph - Fire Journal Science and Technology (accepted). 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), 233 – 242. M.Manjuri and B.Maheswari - Strong Dominating Sets of Strong Product Graph of Cayley Graphs with Arithmetic Graphs - International Journal of Computer Engineering & Technology, Volume 4, Issue 6 (2013), 136 - 144. 11