Journal of Xi'an University of Architecture & Technology ISSN No : 1006-7930 Extended Roman Domination of Product Related Graphs and Star Related Graphs. Dr.Manjula. C. Gudgeri1, Professor 1Department of Mathematics KLE’s Dr. MSSCET, Belagavi, 590008, Karnataka, INDIA Mrs. Varsha2*, Research Scholar 2Department of Mathematics KLE’s Dr. MSSCET, Belagavi, 590008, Karnataka , INDIA Mrs. Pallavi Sangolli3 3Department of Mathematics SGBIT, Belagavi, Karnataka, INDIA Abstract: “An Extended Roman Domination function on a graph G=(V,E) is a function 퐟 ∶ 퐕 → {ퟎ, ퟏ, ퟐ, ퟑ} satisfying the conditions that (i) every vertex 퐮 for which 퐟(퐮) is either ퟎ or ퟏ is adjacent to at least one vertex 퐯 for which 퐟(퐯) = ퟑ, (ii) if 퐮 and 퐯 are two adjacent vertices and if 퐟(퐮) = ퟎ then 퐟(퐯) ≠ ퟎ. The weight of an Extended Roman Domination function is the sum of values assigned to all vertices”. [3], [4], [9] “The minimum weight of an Extended Roman Domination function on a graph G is called the Extended Roman Domination number of G, denoted by 후퐞퐫퐝” [3], [4], [9]. In this paper we study the Extended Roman Domination of Prism, M¨obius ladder, Mongolian Tent, Ladder, Star, Firecracker Graph, Book graph and Bistar graph. AMS Subject Classification: 05C69, 05C78 Keywords: Roman Domination function, Roman Domination number, Extended Roman Domination function, Extended Roman Domination number. I. INTRODUCTION Cockayne et al. [5] (2004) defined “a Roman dominating function (RDF) on a graph G(V, E) to be a function 푓: 푉 → {0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v) =2. The weight of a Roman Dominating function is the value f(V)=∑푢∈푉 푓(푢). The minimum weight of a Roman Dominating function on a graph G is called the Roman Domination number of G, denoted by 훾푟푑”. [1], [4],[9],[11] The “definition of a Roman dominating function was motivated by an article in Scientific American by Ian Stewart entitled “Defend the Roman Empire” [13] and suggested even earlier by ReVelle (1997). Each vertex in our graph represents a location in the Roman Empire. A location (vertex v) is considered unsecured if no legions are stationed there (i.e., 푓(푣) = 0) and secured otherwise (i.e., if f(v) ∈ {1, 2}). An unsecured location (vertex 푣) can be secured by sending a legion to 푣 from an adjacent location (an adjacent vertex 푢). But Constantine the Great (Emperor of Rome) issued a decree in the 4th century A.D. for the defense of his cities. He decreed that a legion cannot be sent from a secured location to an unsecured location if doing so leaves that location unsecured. Thus, two legions must be stationed at a location Volume XII, Issue IX, 2020 Page No: 480 Journal of Xi'an University of Architecture & Technology ISSN No : 1006-7930 ( 푒 푓(푣) = 2) before one of the legions can be sent to an adjacent location. In this way, Emperor Constantine the Great can defend the Roman Empire”. [1], [4],[9] II. DEFINITIONS “An Extended Roman Domination function (ERD) on a graph G=(V,E) is a function 푓: 푉 → {0,1,2,3} satisfying the conditions that (i) every vertex 푢 for which 푓(푢) is either 0 or 1 is adjacent to at least one vertex 푣 for which 푓(푣) = 3, (ii) if 푢 and 푣 are two adjacent vertices and if 푓(푢) = 0 then 푓(푣) ≠ 0. The weight of an Extended Roman Domination function is the value 푓(푣) = ∑푢∈푣 푓(푢). The minimum weight of an Extended Roman Domination function on a graph G is called the Extended Roman Domination number of G, denoted by 훾푒푟푑”. [3],[4],[9] “For a graph G=(V,E), let 푓: 푉 → {0,1,2,3}, and let {푉0,푉1,푉2, 푉3} be the ordered partition of 푉 induced by f, where 푉푖={푣 ∈ 푉 |푓(푣) = } and |푉푖| = 푛푖, for i= 0,1,2,3. Note that there exists a 1-1 correspondence between the functions 푓: 푉 → {0,1,2,3} and the ordered partitions {푉0,푉1,푉2, 푉3} of V. thus we will write 푓 = {푉0,푉1,푉2, 푉3} ”. [3],[4],[5], [6],[9] “A function 푓 = {푉0,푉1,푉2, 푉3} is an Extended Roman Domination function if, (i) 푉3 ≻ 푉0 ∪ 푉1, where ≻ means that the set 푉3 dominates the set 푉0 ∪ 푉1, i.e., 푉0 ∪ 푉1 ⊆ N [푉3] .The weight of 푓 is 푓(푉) = ∑푢∈푣 푓(푢) = 3푛3 + 2푛2 + 푛1 . We say a function 푓 = {푉0,푉1,푉2, 푉3} is a 훾푒푟푑 function if it is an extended Roman Domination function and 푓(푉) = 훾푒푟푑(G)”. [3],[4],[5], [6],[9] III. RESULTS ON EXTENDED ROMAN DOMINATION 3.1 Prism Graph “A generalized prism graph Ym,n is the graph Cartesian product Ym,n= Cn Pm, has mn vertices and m(2n-1)”. [10] Theorem 1: For the Prism graph Ym,n Extended Roman Domination number 9푛+5 , 푓표푟 푛 = 2k + 1, where k = 1,3,5,7, … … . 4 9푛+3 , 푓표푟 n = 2k + 3, where k = 1,3,5,7, … 4 훾푒푟푑 = 9푛 , 푓표푟 푛 = 2푘 + 2, 푤ℎ푒푟푒 푘 = 1,3,5,7, … … 4 9푛+2 , 푓표푟 푛 = 2푘 + 4, 푤ℎ푒푟푒 푘 = 1,3,5,7, … … … . { 4 Proof: Case (i): 푓표푟 푛 = 2k + 1 , where k = 1,3,5,7, … …. Here vertices of the Prism graph Ym,n are labeled as 3, 1 and 0 to obtain the minimum weight. The vertices which 푛+1 푛+1 3푛−1 3푛−1 are labeled 3 are ( ) i.e., |푉 | = |( )| , 1 are ( ) , 푒. , |푉 | = |( )| and rest of the vertices are zero. 2 3 2 4 1 4 3푛−1 3(푛+1) 9푛+5 Therefore the weight of the ERD of Prism graph Ym,n is, |푉 | + 3|푉 | = ( ) + ( ) = ( ), this being the 1 3 4 2 4 9푛+5 minimum weight. Hence 훾 (Ym,n) = ( ) . 푒푟푑 4 Case (ii): 푓표푟 푛 = 2k + 3 where k = 1,3,5,7, … …. Volume XII, Issue IX, 2020 Page No: 481 Journal of Xi'an University of Architecture & Technology ISSN No : 1006-7930 Here vertices of the Prism graph (Ym,n) are labeled as 3, 2,1 and 0 to obtain the minimum weight. The vertices which (푛−1) (푛−1) 3푛−7 3푛−7 are labeled 3 are ( ) i.e., |푉 | = | | , 2 are 2 , 푒. , |푉 | = |2| , 1 are ( ) , 푒. , |푉 | = | ( ) | and rest 2 3 2 2 4 1 4 3푛−7 of the vertices are zero. Therefore the weight of the ERD of Prism graph Ym,n is, |푉 | + 2|푉 | + 3|푉 | = ( ) + 1 2 3 4 3(푛−1) 9푛+3 9푛+3 4 + ( ) = ( ), this being the minimum weight. Hence 훾 (Ym,n) = ( ). 2 4 푒푟푑 4 Case (iii): 푓표푟 푛 = 2k + 2 where k = 1,3,5,7, … …. Here vertices of the Prism graph Ym,n are labeled as 3, 1 and 0 to obtain the minimum weight. The vertices which 푛 푛 3푛 3푛 are labeled 3 are ( ) i.e.,|푉 | = |( )| ,1 are ( ) , 푒. , |푉 | = |( )| and rest of the vertices are zero. Therefore the 2 3 2 4 1 4 3푛 3(푛) 9푛 weight of the ERD of Prism graph Ym,n is, |푉 | + 3|푉 | = ( ) + ( ) = ( ), this being the minimum weight. 1 3 4 2 4 9푛 Hence 훾 (Ym,n) = ( ) . 푒푟푑 4 Case (iv): 푓표푟 푛 = 2k + 4 where k = 1,3,5,7, … …. Here vertices of the Prism graph Ym,n are labeled as 3, 2,1 and 0 to obtain the minimum weight. The vertices which 푛 푛 3(푛−2) 3(푛−2) are labeled 3 are ( ) i.e., |푉 | = |( )| , 2 are 1 , 푒. , |푉 | = |1|, 1 are ( ) , 푒. , |푉 | = |( )| , and rest of 2 3 2 2 4 1 4 3(푛−2) the vertices are zero. Therefore the weight of the ERD of Prism graph Ym,n is, |푉 | + 2|푉 | + 3|푉 | = ( ) + 1 2 3 4 3(푛) 9푛+2 9푛+2 2 + ( ) = ( ), this being the minimum weight. Hence 훾 (Ym,n )= ( ) . 2 4 푒푟푑 4 3.2 M¨obius ladder “The M¨obius ladder Mn is the graph obtained from the ladder Pn ×P2 by joining the opposite end points of the two copies of Pn”. [8] Theorem 2: For the M¨obius ladder graph Mn Extended Roman Domination number 9푛+1 , 푓표푟 푛 = 2k + 1 where k = 1,3,5,7, … … . 4 9푛+3 , 푓표푟 n = 2k + 3, where k = 1,3,5,7, … 4 훾푒푟푑 = 9푛+4 , 푓표푟 푛 = 2푘 + 2, 푤ℎ푒푟푒 푘 = 1,3,5,7, … … 4 9푛+6 , 푓표푟 푛 = 2푘 + 4, 푤ℎ푒푟푒 푘 = 1,3,5,7, … … … . { 4 Proof: Case (i): 푓표푟 푛 = 2k + 1 where k = 1,3,5,7, … …. Here vertices of the M¨obius ladder graph Mn are labeled as 3,2, 1 and 0 to obtain the minimum weight. The 푛−1 푛−1 3푛−9 vertices which are labeled 3 are ( ) i.e., |푉 | = |( )| , 2 are 2 i.e., |푉 | = |2| ,1 are ( ) , 푒. , |푉 | = 2 3 2 2 4 1 3푛−9 |( )| and rest of the vertices are zero. Therefore the weight of the ERD of M¨obius ladder graph Mn is , |푉 | + 4 1 3푛−9 3(푛−1) 9푛+1 2|푉 | + 3|푉 | = ( ) + 4 + ( ) = ( ), this being the minimum weight. 2 3 4 2 4 9푛+1 Hence 훾 (Mn) = ( ) . 푒푟푑 4 Case (ii): 푓표푟 푛 = 2k + 3 where k = 1,3,5,7, … …. Here vertices of the M¨obius ladder (Mn) are labeled as 3, 2,1 and 0 to obtain the minimum weight.