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50120140501003
1. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING & 6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME TECHNOLOGY (IJCET) ISSN 0976 – 6367(Print) ISSN 0976 – 6375(Online) Volume 5, Issue 1, January (2014), pp. 21-27 © IAEME: www.iaeme.com/ijcet.asp Journal Impact Factor (2013): 6.1302 (Calculated by GISI) www.jifactor.com IJCET ©IAEME ROMAN DOMINATING AND TOTAL DOMINATING FUNCTIONS OF CORONA PRODUCT GRAPH OF A PATH WITH A STAR 1 Makke Siva Parvathi, 2 Bommireddy Maheswari 1 2 Dept. of Mathematics, K.R.K. Govt. Degree College, Addanki-523201, Andhra Pradesh, India Dept. of Applied Mathematics, S.P.Women’s University, Tirupat-517502, Andhra Pradesh, India ABSTRACT In recent years ‘Domination in Graphs’ received much attention and it is an important branch of graph theory. Haynes et al. [1, 2] have given an extensive overview on domination in graphs and related topics in the two books. Recently dominating functions in domination theory have received much attention. In this paper we present some results on minimal Roman and total Roman dominating functions of corona product graph of a path with a star. Key Words: Corona Product, Roman dominating function, Total Roman dominating function. Subject Classification: 68R10 1. INTRODUCTION Graph theory is one of the most flourishing branches of modern mathematics and finds application in various branches of Science & Technology. Domination in graphs has a wide range of applications to many fields like Engineering, Communication Networks, Social sciences, linguistics, physical sciences and many others. Allan, R.B. and Laskar, R.[3], Cockayne, E.J. and Hedetniemi, S.T. [4] have studied various domination parameters of graphs. The Roman dominating function of a graph G was defined by Cockayne et al. [5]. The definition of a Roman dominating function was motivated by an article in Scientific American by Ian Stewart [6] entitled “Defend the Roman Empire!” and suggested by even earlier by ReVelle[7]. Frucht and Harary [8] 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. 21
2.
International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME The authors have studied some dominating functions of corona product graph of a cycle with a complete graph [9] and published papers on minimal dominating functions, some variations of Y – dominating functions, Y – total dominating functions , total dominating functions and convexity of dominating and total dominating functions[10,11,12,13,14]. In this paper we discuss some results on Roman dominating functions and total Roman dominating functions of corona product graph of a path with a star. 2. CORONA PRODUCT OF AND , 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 require the following theorem whose proof can be found in Siva Parvathi, M. [8]. Theorem 2.1: 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 3. ROMAN DOMINATING FUNCTIONS In this section we discuss Roman dominating functions and minimal Roman dominating functions of the graph ܩൌ ܲ Í ܭଵ, . First we recall the definitions of Roman dominating function of a graph. Definition: Let ܩሺ ܸ, ܧሻ be a graph. A function ݂ ܸ ՜ ሼ 0, 1, 2 ሽ is called a Roman dominating function (RDF) of ܩif ݂ሺܰሾݒሿሻ ൌ ∑ f (u ) ≥ 1, ݂݄ܿܽ݁ ݎ ܸ ∈ ݒand satisfying the condition that every vertex ݑfor which ݂ሺݑሻ ൌ 0 is adjacent to at least one vertex ݒfor which ݂ ሺݒሻ ൌ 2. A Roman dominating function ݂ of ܩis called a minimal Roman dominating function (MRDF) if for all ݃ ൏ ݂, ݃ is not a Roman dominating function. Theorem 3.1: A function ݂ ܸ ՜ ሼ 0, 1, 2 ሽ defined by 2, for v whose degree is m + 1 in each copy of K 1, m in G, ݂ ሺ ݒሻ ൌ 0, otherwise. is a minimal Roman dominating function of ܩൌ ܲ Íܭଵ, . Proof: Let ݂ be a function defined as in the hypothesis. Case 1: Let ܲ א ݒbe such that ݀ ሺ ݒሻ ൌ ݉ 3 in .ܩ Then ܰሾݒሿ contains ݉ 1 vertices of ܭଵ, and three vertices of ܲ in .ܩ So ∑ f (u ) = 0 + 0 + 0 + 2 + 0 + ....... + 0 = 2. 14 4 2 3 u∈N [v ] m −times Case 2: Let ܲ א ݒbe such that ݀ ሺ ݒሻ ൌ ݉ 2 in .ܩ Then ܰሾݒሿ contains ݉ 1 vertices of ܭଵ, and two vertices of ܲ in .ܩ So ∑ f (u ) = 0 + 0 + 2 + 0 + ....... + 0 = 2. 14 4 2 3 u ∈ N [v ] m −times Case 3: Let ܭ א ݒଵ, be such that ݀ሺݒሻ ൌ ݉ 1 in .ܩ u∈N [v ] 22
3.
International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME Then ܰሾݒሿ contains ݉ 1 vertices of ܭଵ, and one vertex of ܲ in .ܩ So ∑ f (u ) = 0 + 2 + 0 + ....... + 0 = 2. 14 4 2 3 u ∈ N [v ] m −times Case 4: Let ܭ א ݒଵ, be such that ݀ሺݒሻ ൌ 2 in .ܩ Then ܰሾݒሿ contains two vertices of ܭଵ, and one vertex of ܲ in .ܩ So ∑ f (u ) = 0 + 2 + 0 = 2. u∈N [v ] Therefore for all possibilities, we get ∑ f (u ) > 1, ∀ v ∈ V. Let ݑbe any vertex in ܩsuch that ݂ሺݑሻ ൌ 0. Then ܲ ∈ ݑor ܭ ∈ ݑଵ, such that ݀ሺݑሻ ൌ 2. Let ݑ ് ݒbe a vertex in ܩsuch that ݂ሺݒሻ ൌ 2. Then ܭ ∈ݒଵ, such that ݀ሺݒሻ ൌ ݉ 1. We now show that ݑis adjacent to . ݒIf ܲ ∈ ݑ , then ݑis adjacent to , ݒsince every vertex in ܲ is adjacent to every vertex in the corresponding copy of ܭଵ, . Also if ܭ ∈ ݑଵ, then ݑis adjacent to ܭ ∈ ݒଵ, , since ݑand ݒare in different partitions of ܭଵ, . This implies that ݂ is a RDF. Now we check for the minimality of ݂. Define g : V → {0, 1, 2} by u∈N [v ] 1, ݃ሺ ݒሻ ൌ 2, 0, for v whose degree is m + 1 in the i th copy of K 1,m in G, for v whose degree is m + 1 in (n - 1) copies of K 1, m in G, otherwise. Since strict inequality follows at the vertex v ∈ V , it follows that ݃ ൏ ݂. Case (i): Let ܲ א ݒbe such that ݀ሺݒሻ ൌ ݉ 3 in .ܩ Sub case 1: Let ݒbe in the ݅ ௧ copy of ܭଵ, in .ܩ Then ∑ g (u ) = 0 + 0 + 0 + 1 + 0 + ....... + 0 = 1. 14 4 2 3 u∈N [v ] m −times Sub case 2: Let ݒbe not in the ݅ ௧ copy of ܭଵ, in .ܩ Then ∑ g (u ) = 0 + 0 + 0 + 2 + 0 + ....... + 0 = 2. 14 4 2 3 u ∈ N [v ] m − times Case (ii): Let ܲ א ݒ be such that ݀ሺ ݒሻ ൌ ݉ 2 in .ܩ Sub case 1: Let ݒbe in the ݅ ௧ copy of ܭଵ, in .ܩ Then ∑ g (u ) = 0 + 0 + 1 + 0 + ....... + 0 = 1. 14 4 2 3 u ∈ N [v ] m − times Sub case 2: Let ݒbe not in the ݅ ௧ copy of ܭଵ, in .ܩ Then ∑ g (u ) = 0 + 0 + 2 + 0 + ....... + 0 = 2. 14 4 2 3 u∈N [v ] m −times ሺݒሻ ൌ ݉ 1 in .ܩ Case (iii): Let ܭ א ݒଵ, be such that ݀ Sub case 1: Let ݒbe in the ݅ ௧ copy of ܭଵ, in .ܩ Then ∑ g (u ) = 0 + 1 + 0 + ....... + 0 = 1. 14 4 2 3 u ∈ N [v ] m − times ௧ Sub case 2: Let ݒbe not in the ݅ copy of ܭଵ, in .ܩ 23
4.
International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME Then ∑[ ] g (u ) = 0 + 2 + 0 +4243 = 2. 1 ....... + 0 Case (iv): Let ܭ א ݒଵ, be such that ݀ሺݒሻ ൌ 2 in .ܩ Sub case 1: Let ݒbe in the ݅ ௧ copy of ܭଵ, in .ܩ Then ∑ g (u ) = 0 + 1 + 0 = 1. u∈ N v m − times u∈N [v ] Sub case 2: Let ݒbe not in the ݅ ௧ copy of ܭଵ, in . ܩ Then ∑ g (u ) = 0 + 2 + 0 = 2. u∈N [v ] This implies that ∑ g (u ) ≥ 1, ∀ v ∈ V. u∈N [v ] i.e. ݃ is a DF. But ݃ is not a RDF, since the RDF definition fails in the ݅ ௧ copy of ܭଵ, in . ܩ Because the vertex ݑin the ݅ ௧ copy of ܭଵ, of ܩfor which ݂ሺݑሻ ൌ 0 is adjacent to a vertex ݒfor which ݂ ሺݒሻ ൌ 1. Therefore ݂ is a MRDF. ז 4. TOTAL ROMAN DOMINATING FUNCTIONS In this section we introduce the concept of total Roman dominating function of a graph G. Some results on minimal total Roman dominating function of ܩൌ ܲ ܭଵ, are obtained. First we define total Roman dominating function of a graph. Definition: Let ܩሺ ܸ, ܧሻ be a graph. A function ݂ ܸ ՜ ሼ 0, 1, 2ሽ is called a total Roman dominating function (TRDF) of ܩif ݂ ሺܰሺݒሻሻ ൌ ∑ f (u ) ≥ 1, ݂ ܸ ∈ ݒ ݄ܿܽ݁ ݎand satisfying u∈N ( v ) the condition that every vertex ݑfor which ݂ሺݑሻ ൌ 0 is adjacent to at least one vertex ݒfor which ݂ ሺݒሻ ൌ 2. A total Roman dominating function ݂ of ܩis called a minimal total Roman dominating function (MTRDF) if for all ݃ ൏ ݂, ݃ is not a total Roman dominating function. Theorem 4.1: A function f : V → { 0, 1, 2} defined by 2, for the vertices of Pn in G, ݂ሺݒሻ ൌ 0, otherwise . is a minimal Total Roman Dominating Function of ܩൌ ܲ ܭଵ, . Proof: Let ݂ be a function defined as in the hypothesis. Case 1: Let ܲ א ݒbe such that ݀ ሺ ݒሻ ൌ ݉ 3 in .ܩ Then ܰሺݒሻ contains ݉ 1 vertices of ܭଵ, and two vertices of ܲ in .ܩ So ∑ f (u ) = 2 + 2 + 0 + 04........ + 0 = 4. 14 +244 3 u∈N ( v ) ( m +1) −times Case 2: Let ܲ א ݒbe such that ݀ ሺ ݒሻ ൌ ݉ 2 in .ܩ Then ܰሺݒሻ contains ݉ 1 vertices of ܭଵ, and one vertex of ܲ in .ܩ So ∑ f (u ) = 2 + 0 + 04........ + 0 = 2. +244 14 3 u∈N ( v ) ( m +1) −times ሺݒሻ ൌ ݉ 1 in .ܩ Case 3: Let ܭ א ݒଵ, be such that ݀ Then ܰሺݒሻ contains ݉ vertices of ܭଵ, whose degree is 2 and one vertex of ܲ in .ܩ 24
5.
International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME So ∑ f (u ) = 2 + 0 +44244 0 = 2. 1 0 + ........ + 3 m −times Case 4: Let ܭ א ݒଵ, be such that ݀ሺݒሻ ൌ 2 in .ܩ Then ܰሺݒሻ contains one vertex of ܭଵ, whose degree is ݉ 1 and one vertex of ܲ in .ܩ So ∑ f (u ) = 2 + 0 = 2. u∈N ( v ) u∈N ( v ) ∑ f (u ) > 1, Therefore ∀ v ∈ V. u∈N ( v ) And every vertex ݑfor which ݂ ሺ ݑሻ ൌ 0 is in ܭଵ, and every such vertex is adjacent to a corresponding vertex say ݒin ܲ for which ݂ሺݒሻ ൌ 2. This implies that ݂ is a TRDF. Now we check for the minimality of ݂ . Define g : V → { 0, 1, 2 } by 1, for v = v k ∈ Pn in G, ݃ሺ ݒሻ ൌ 2, for v ∈ Pn - {v k } in G, 0, otherwise. Case (i): Let ܲ א ݒbe such that ݀ሺݒሻ ൌ ݉ 3 in G. vk ∈ N (v) . Sub case 1: Let Then ∑ g (u ) = 1 + 2 + 0 +44244 0 = 3. 1 0 + ........ + 3 u∈N ( v ) ( m+1)−times Sub case 2: Let vk ∉ N (v) . Then ∑ g (u ) = 2 + 2 + 0 +44244 0 = 4. 1 0 + ........ + 3 (m +1)−times Case (ii): Let ܲ א ݒ be such that ݀ሺ ݒሻ ൌ ݉ 2 in G. Sub case 1: Let vk ∈ N (v) . u∈N ( v ) Then ∑ g (u ) = 1 + 0 +44244 0 = 1. 1 0 + ........ + 3 (m +1)−times Sub case 2: Let vk ∉ N (v) . u∈N ( v ) Then ∑ g (u ) = 2 + 0 +44244 0 = 2. 1 0 + ........ + 3 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 ) = 1 + 0 +44244 0 = 1. 1 0 + ........ + 3 u∈N ( v ) m −times Sub case 2: Let vk ∉ N (v) . Then ∑ g (u ) = 2 + 0 +44244 0 = 2. 1 0 + ........ + 3 m − times Case (iv): Let ܭ א ݒଵ, be such that ݀ ሺ ݒሻ ൌ 2 in G. u∈N ( v ) 25
6.
International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME Sub case 1: Let vk ∈ N (v) . Then ∑ g (u ) = 1 + 0 = 1. u∈N ( v ) Sub case 2: Let vk ∉ N (v) . Then ∑ g (u ) = 2 + 0 = 2. ∀ v ∈ V. u∈N ( v ) Therefore for all possibilities, we get ∑ g (u ) ≥ 1, i.e. ݃ is a TDF. But ݃ is not a TRDF, since the TRDF definition fails in the ݇ ௧ copy of ܭଵ, in G because the vertex ݑin the ݇ ௧ copy of ܭଵ, of G for which ݂ሺݑሻ ൌ 0 is adjacent to a vertex ݒ for which ݂ ሺݒ ሻ ൌ 1. Therefore ݂ is a MTRDF. ז u∈N ( v ) 5. ILLUSTRATION 0 0 2 0 0 0 2 2 0 0 0 0 2 2 0 0 0 0 0 0 0 0 2 0 0 2 2 0 0 0 0 ൌ ૡ ٖ , 0 The function ‘f’ takes the value 2 for the vertices of ૡ and the value 0 for the remaining vertices in G. 6. 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] Allan, R.B. and Laskar, R.C. (1978) – On domination, independent domination numbers of a graph Discrete Math., 23, pp.73 – 76. 26
7.
International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] Cockayne, E.J. and Hedetniemi, S.T. (1977) - Towards a theory of domination in graphs. Networks, 7, pp.247 – 261. Cockayne, E.J., Dreyer, P.A., Hedetniemi, S.M. and Hedetniemi, S.T. (2004) - Roman domination in graphs, Discrete Math., 278, 11 – 22. Ian Stewart (1999) - Defend the Roman Empire!, Scientific American, 281 (6), 136 – 139. ReVelle, C.S. and Rosing, K.E. (2000) - Defendens imperium romanum: a classical problem in military strategy, Amer.Math.Monthly, 107(7), 585 – 594. Frucht, R. and Harary, F. (1970) - On the corona of Two Graphs, Aequationes Mathematicae, Volume 4, Issue 3, pp.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, pp. 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, pp. 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). Siva Parvathi, M and Maheswari, B. - Minimal total dominating functions of corona product graph of a cycle with a complete graph – International Journal of Applied Information Systems (communicated). Siva Parvathi, M and Maheswari, B. - Convexity of minimal dominating and total dominating functions of corona product graph of a cycle with a complete graph – International Journal of Computer Applications(communicated). 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. 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), pp. 136 - 144. 27
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