Turkish Journal of Computer and Mathematics Education Vol.12 No.11 (2021), 2166-2181 Research Article Reserved Domination Number of some Graphs Dr. G. Rajasekar1, G. Rajasekar2 1Associate Professor, PG and Research Department of Mathematics, Jawahar Science College, Neyveli 2Research Scholar, PG and Research Department of Mathematics, Jawahar Science College, Neyveli [email protected]., [email protected] Article History: Received: 10 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021; Published online: 10 May 2021 Abstract: In this paper the definitions of Reserved domination number and 2-reserved domination number are introduced as for the graph GVE= ( , ) a subset S of V is called a Reserved Dominating Set (RDS ) of G if (i) R be any nonempty proper subset of S ; (ii) Every vertex in VS− is adjacent to a vertex in S . The dominating set S is called a minimal reserved dominating set if no proper subset of S containing R is a dominating set. The set R is called Reserved set. The minimum cardinality of a reserved dominating set of G is called the reserved domination number of G and is denoted by where is the number of reserved vertices. Using these definitions the 2-reserved domination number for RG(k) − ( ) k Path graph Pn , Cycle graph Cn , Wheel graph Wn , Star graph Sn , Fan graph F1,n , Complete graph Kn and Complete Bipartite graph Kmn, are found. Keywords: Dominating set, Reserved dominating set, 2-Reserved dominating set. 1. Introduction (Times New Roman 10 Bold) Domination in graphs has wide applications to several fields such as mobile Tower installation, School Bus Routing, Computer Communication Networks, Radio Stations, Locating Radar Stations Problem, Nuclear Power Plants Problem, Modeling Biological Networks, Modeling Social Networks, Facility Location Problems, Coding Theory and Multiple Domination Problems with hierarchical overlay networks. Domination arises in facility location problems, where the number of facilities (e.g., Mobile towers, bus stop, primary health center, hospitals, schools, post office, hospitals, fire stations) is fixed and one attempts to minimize the distance that a person needs to travel to get to the nearest facility. A similar problem occurs when the maximum distance to a facility is fixed and government or any service provider attempts to minimize the number of facilities necessary so that everyone is serviced. Concepts from domination also appear in problems involving finding sets of representatives, in monitoring communication or electrical networks, and in land surveying. Let GVE= ( , ) be a graph. A subset S of V is called a dominating set of G if every vertex in VS− is adjacent to a vertex in S . A dominating set S is called a minimal dominating set if no proper subset of S is a dominating set. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by (G) . The maximum cardinality of a minimal dominating set of G is called the upper domination number of G and is denoted by (G) . Consider the situation of installing minimum number of Mobile phone towers so that it will be utilized by all the people living in towns or villages. Here each town or village is considered as separate entity called as vertex. These towns and villages are connected by roads (In this case Ariel distance) called edges. The situation of installing minimum number of Mobile phone Towers is domination problem and this minimum number is domination number. In real life situation it is not always in practice of installing the Towers only at the public utility pattern. If the installing authorities are very much interested in installing the tower, nearer to their residence or nearer to the residence of VIPs or to their favorites’ residence then the concerned person will reserve some of the installation points without any concern about the nearer or proximity of the other towns or villages. Such type installation points are a called reserved installation points. This situation motives the development of the reserved domination points. These reserved points are automatically included in the domination set. 2166 Turkish Journal of Computer and Mathematics Education Vol.12 No.11 (2021), 2166-2181 Research Article 2.Resaserved domination number Definition: Dominating Set Let GVE= ( , ) be a graph. A subset S of V is called a dominating set of G if every vertex in VS\ is adjacent to a vertex in S . A dominating set S is called a minimal dominating set if no proper subset of S is a dominating set. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by (G) . The maximum cardinality of a minimal dominating set of G is called the upper domination number of G and is denoted by (G) . S V G R Definition: Reserved Domination set S VS− S S Let GVE= ( , ) be a graph. A subset of is called a Reserved Dominating Set (RDS ) of if S R R (i) be any nonempty proper subset of . G G (ii) Every vertex ink is adjacent to a vertex in . The dominating set is called a minimal reserved dominating set if no proper subset of containing is a dominating set. The set is called Reserved set. The minimum cardinality of a reserved dominating set of is called the reserved domination number of and is denoted by RG(k) − ( ) where is the number of reserved vertex. Remark: Then , we get the 2-reserved domination number of and is denoted by . In this k = 2 G RG(2) − () paper let us find the 2-reserved domination number of various graphs. Definition: Graph Connected by a Bridge Let G , G be any two graphs and G be a graph attained by connecting G and G by a bridge e= v v 1 2 1 2 12 where v11 V( G ) , v22 V( G ) . 3. Some Preliminaries results Domination Number: n + 2 (i) Lollipop Graph: (Lmn, ) = . 3 mn++21 (ii) Tadpole Graph: (Tmn, ) =+. 33 n (iii) Pan Graph: (Pn,1 ) = . 3 (iv) Barbell Graph: (Bn ) = 2 . Definition: Lollipop Graph The lollipop graph is the graph obtained by joining a complete graph Km with m 3 to a path graph Pn with a bridge. It is denoted by Lmn, . Theorem: For the lollipop graph Lmn, , the reserved domination number is 2167 Turkish Journal of Computer and Mathematics Education Vol.12 No.11 (2021), 2166-2181 Research Article −n 1 1+= , if u1 3 n RL−=, 1+ , if =u, k = 2,3,4,..., m (1) ( mn, ) 3 k k − 3 nk−+( 1) 2+ + , if =vk , k = 1,2,3,..., n 33 Proof: Fig 1: A Lollipop Graph Lmn, . For convenience let us consider the lollipop graph Lmn, into two entities. One is complete graph Km with vertices u1, u 2 , u 3 ,..., um and another one is the path graph P with verticesv1, v 2 , v 3 ,..., vn . n Case (i): Suppose u1 is the reserved vertex. Then u1 must be in the dominating set and dominates the verticesu2, u 3 , u 4 ,..., um v 1 . Now it is enough to find the dominating set for the remaining vertices v23, v ,..., vn . The LVmn,1 with V1= v 2, v 3 ,..., vn is nothing but Pn−1 . Hence R(1) −( Lm, n,1 u 1) = + ( P n− 1 ) n −1 =+1 . 3 Case (ii): Suppose uk , k= 2,3,4,..., m is the reserved vertex. uk Dominates the vertices u1, u 2 ,..., uk−+ 1 , u k 1 ,..., u m and doesn’t dominate the vertices v1, v 2 , v 3 ,..., vn . Since the LVmn,2 with V2= v 1, v 2 , v 3 ,..., vn is Pn , RLP(1) −( m, n,1 ) = + ( n ) n =+1 , where ==uk , k 2,3,..., m . 3 Case (iii): Suppose vk , k= 1,2,3,..., n is the reserved vertex. To dominate the set it is enough to choose anyone of the vertex from that set. But if we choose u1 alone then it would dominate the vertex u23, u ,..., um as well asv1 . So u1 must be in the required dominating set. Since vk is the reserved vertex, it dominates vk−1 and vk+1 . Now the vertices which are not dominated while considering the set vuk , 1 as a subset of the dominating set are U= v2, v 3 ,..., vk−+ 2 , v k 2 ,..., v n . 2168 Turkish Journal of Computer and Mathematics Education Vol.12 No.11 (2021), 2166-2181 Research Article LUPPm,3 n = k− nk−+( 1) . (LUPPm,3 n ) = ( k− ) + ( nk−+( 1) ) Hence R(11) −( Lm, n,, ) = u 1 v k + ( P k− 3 ) + ( Pnk−+( ) ) k −3 nk−+( 1) =2 + + , where . ==vk , k 1,2,3,..., n 33 Definition: Tadpole Graph: The (mn, ) Tadpole graph, also called dragon graph denoted by Tmn, is the graph obtained by joining a cycle graph Cm to a path graph Pn with a bridge. Theorem: For the Tadpole graph Tmn, , the reserved domination number is 2169 Turkish Journal of Computer and Mathematics Education Vol.12 No.11 (2021), 2166-2181 Research Article mn −1 uk ( k=1,2,3,..., m) +=, if 33 v k=−2,3,5,..., n 1, n k ( ) for mn, 0 mod3 ( ) mn−11 + + , if =vk ( k =1,4,7,..., n − 5, n − 2) 33 mn −1 ukk ( =1,4,7,...,mm−−5, 2) +=, if 33 vk ( k=3,6,9,..., n − 4, n − 1) for mn0( mod3) , 1( mod3) mn uk ( k=−2,3,5,..., m 1, m) +=, if 33 v k=−1,2,4,..., n 2, n k ( ) mn uk ( k=1,2,3,..., m) +=, if for mn0( mod3) , 2( mod3) 33 vkk ( =1,2,3,...,n) mn uk ( k=1,2,3,..., m) +, if = for mn 1( mod3) , 0( mod3) 33 vk ( k=1,2,3,..., n) mn −1 uk ( k=1,2,3,..., m) +=, if 33 vkk ( =−1,3,4,...,nn 1, ) RT(1) −=( mn, , ) for mn, 1( mod3) mn +, if =vk ( k =2,5,8,..., n − 5, n − 2) 33 u1, u 2 , u 3 ,..., um Pn v1, v 2 , v 3 ,..., vn mn −1 uk ( k=−1,2,4,..., m 2, m) +=, if 33 vk ( k=−2,3,5,..., n 2, n) for mn1( mod3) , 2( mod3) mn−+11 uk ( k=3,6,9,..., m − 4, m − 1) + , if = 33 v k=1,4,7,..., n − 4, n − 1 k ( ) mn −1 uk ( k=1,2,3,..., m) += , if 33 vk ( k=−2,3,5,..., n 1, n) for mn2( mod3) , 0( mod3) mn−+11 + , if =vk ( k =1,4,7,..., n − 5, n − 2) 33 mn −1 uk ( k=1,3,4,..., m − 2, m − 1) +=, if 33 vk ( k=3,6,9,..., n − 4, n − 1) for m 2 mod3 ,n 1 mod3 ( ) ( ) mn uk ( k=−2,5,8,..., m 3, m) +=, if 33 vk ( k=−1,2,4,..., n 2, n) mn uk ( k=1,2,3,..., m) +, if = for mn, 2(mod3) 33 v k=1,2,3,..., n k ( ) Proof: For convenience let us consider the tadpole graph Tmn, into two entities.