Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase Volume 4, Number 1, 2023, pp.30-39 https://doi.org/10.5206/mase/15775 MINIMUM DOMINATING SET FOR THE PRISM GRAPH FAMILY JEBISHA ESTHER S AND VENINSTINE VIVIK J Abstract. The dominating set of the graph G is a subset D of vertex set V , such that every vertex not in V −D is adjacent to at least one vertex in the vertex subset D. A dominating set D is a minimal dominating set if no proper subset of D is a dominating set. The number of elements in such set is called as domination number of graph and is denoted by γ(G). In this work the domination numbers are obtained for family of prism graphs such as prism CLn, antiprism Qn and crossed prism Rn by identifying one of their minimum dominating set. 1. Introduction Domination in graphs is a wide research area in graph theory. The dominating set of the graph G is a subset D of a vertex set V such that every vertex not in V −D is adjacent to at least one vertex in the vertex subset D [6]. A dominating set D is a minimal dominating set if no proper subset of D is a dominating set. The number of elements in such a set is known as the domination number of graph [7]. The basic definition and details about the domination sets and domination numbers of graphs are discussed in [10]. Although the mathematical study of domination in graphs began around 1960, there are some references to domination-related problems about 100 years prior. In 1862, de Jaenisch attempted to determine the minimum number of queens required to cover an n×n chess board. With reference to [9] in many fields such as school bus routing, computer communication networks, radio stations, the locating radar station problem, modeling biological networks, facility location problems and coding theory, the domination is applied. In 2021, Adel et al. [1] have successfully used the Minimum Dominating Sets (MDSets) method to extract proteins that control Protein-Protein Interaction (PPI) networks, revealing a link between structural analysis and biological functions. Motivation Many real-time situations can be modeled as graphs. For instance, each floor in a multi-storey building can be modelled as an n− prism graph [12] by considering the corners of the corridor positions in a floor to be vertices and the side brick walls on both sides as connecting edges. Also, the vertices located in similar at the corner regions of the hallway are connected to each other. This reflects the structure of prism graph. Consider the example of a 4−prism graph obtained from the following floor plan of a rectangular multi-storey building. To strengthen the security system of the building, surveillance cameras may be fixed at various positions. In order to minimize the cost of fixing cameras at various parts of the building, we need some leading positions to cover the whole area. Such dominating points can be visualized using graph- theoretical concept of domination. The minimum number of cameras needed to be installed, which is cost-effective, can be achieved through a minimally dominant set. To handle these types of situations, Received by the editors 20 January 2023; accepted 10 March 2023; published online 22 March 2023. 2020 Mathematics Subject Classification. 05C38, 05C69. Key words and phrases. Minimal dominating set, domination number, prism, antiprism. 30 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15775 MDS FOR PRISM FAMILY OF GRAPH 31 Figure 1. Floor plan of a multi-storey building v1 v2 v3v4 u1 u2 u3u4 Figure 2: Prism Graph CL4 domination in graphs can be applied. Consequently, it motivates us to study the minimal domination of the family of prism graphs. 2. Preliminaries In this section, the literature review and some basic definitions related to our work are given. Here we consider only the graphs that are undirected and cycle-structured. Because of the richness in applications, huge collection of research works has been carried out in this area dealing with domination of graphs. Recently [4] researchers Behzad and et al. studied about an infinite family of regular graphs, the generalized Petersen graphs. The authors [15] in the year 2019 examined the domination of Fibonacci cubes and also they gave the pattern of minimum dominating sets of Fibonacci cubes and in the same year [1] Al-Harere and Breesam investigated the domination number of a new graph called as spinner graph. [3] Alvarado et al. explored about the domination of tree and established the bound for minimal dominating set of forests. In 2021 [11] Liu, Chanjuan discussed about the upper and lower bounds of domination number for n maximal outer planar graph. [5] Bermudo and others proposed some lower bounds on the domination number of a catacondensed hexagonal system using the number of hexagons and the number of branching hexagons[13]. Sarah et al. found certain bounds for the domination number of Latin square graphs. A dominating set [9] for a graph G = (V,E) is a subset D of V such that every vertex not in D is adjacent to at least one number of D. The domination number γ(G) is number of vertices in a smallest dominating set for G. 32 JEBISHA ESTHER S AND VENINSTINE VIVIK J A prism graph [8], denoted by CLn called also as circular ladder graph, is a graph corresponding to the skeleton of an n-prism graph had 2n vertices and 3n edges. An antiprism graph [16] is a graph that has one of the antiprism as its skeleton. An n-sided antiprism graph has 2n vertices and 4n vertices. They are regular, polyhedral. An antiprism graph is a special case of a circulant graph Ci2n(1, 2). Let us denote this graph by Qn. Let n be a positive even integer of at least 4. An n-crossed prism graph [14] is a graph obtained by taking two disjoint cycle graphs on n vertices, namely Cn1 and C n 2 , where V (C n 1 ) = {x1x2, . . . ,xn} and V (Cn2 ) = {w1,w2, . . . ,wn}, such that E(Cn1 ) = (xixi+1,x1xn) where i = 1, 2, . . . ,n − 1 and E(Cn2 ) = (wiwi+1,w1wn) such that i = 1, 2, . . . ,n−1 adding some edges wsxs+1 for s ∈{1, 3, . . . ,n−1} and wtxt−1 for t ∈{2, 4, . . . ,n}. Due to many applications of domination in discrete mathematics, coding theory and networking, etc. It still remains an active research field for many decades. Also it helps in optimal identification of minimum nodes to cover the entire graph. Because of the cyclic structured nature of graphs in this work we have considered this family prism graphs and followed the method of mathematical induction to prove the bounds for domination number. In this paper the domination number for family of prism graphs namely prism graph, antiprism graph and crossed prism graph are obtained. 3. Computation of domination numbers for the prism graph family Theorem 3.1. The domination number of the prism graph CLn, for n ≥ 4, is (1) γ(CLn) = n 2 , if n ≡ 0 mod 4 (2) γ(CLn) = dn2e, if n ≡ 1 mod 4 (3) γ(CLn) = n+2 2 , if n ≡ 2 mod 4 (4) γ(CLn) = dn2e, if n ≡ 3 mod 4 Proof. Let CLn be the prism graph with vertex set V and edge set E. The vertex set is given by V = {vj}∪{uj} where j = 1, 2, . . . ,n. The number of vertices v in CLn is |V | = 2n and the number of edges in CLn is |E| = 3n. One of the minimum dominating set is identified from the following four cases on the various values of n. Case 1. When n ≡ 0 mod 4 the minimum dominating set is given by Dm = {v1,vi}∪{u3,uk} where i = 4p + 1,k = 4p + 3 and p = 1, 2, . . . , n− 4 4 . We used mathematical induction method [17], to prove that given above set is one of the minimal dominating sets of prism graph when n ≡ 0 mod 4. Let n = 4q and q ≥ 1. When q = 1 it gives n = 4 and D1 = {v1,u3} = {vn−3,un−1} Induction Hypothesis on q: Assume that the result is true for the case q = l and then prove that the result is true for q = l + 1. By induction hypothesis the given set Dl = {v1,vi}∪{u3,uk} is the minimal dominating set. Therefore p = 1, 2, . . . , l− 1, i = 5, 9, . . . , 4l− 3 and k = 7, 11, . . . , 4l− 1. =⇒ Dl = {v1,v5,v9, . . . ,v4l−3}∪{u3, . . . ,u4l−1} is true. To prove: When q = l + 1 the result is true. Let q = l + 1 =⇒ n = 4l + 4 =⇒ Dl+1 = {v1,v5, . . . ,v4l+1}∪{u3, . . . ,u4l+3}. Dl+1 = Dl ∪D1 = {v1,v5,v9, . . . ,v4l−3}∪{u3, . . . ,u4l−1}∪{v1,u3}∪{v4l+1,u4l+3}. MDS FOR PRISM FAMILY OF GRAPH 33 This proves the result is true for q = l + 1. =⇒ |Dm| = n 2 Therefore γ(CLn) = n 2 . Case 2. If n ≡ 1 mod 4 then Dm = {v1,vi,vn}∪{u3,uk}where i = 4p + 1, k = 4p + 3 and p = 1, 2, . . . , n− 5 4 . Let n = 4q + 1 and q ≥ 1. Assume q = 1 =⇒ n = 5 =⇒ D1 = {v1,v5,u3} = {v1,vn,un−2} Induction Hypothesis on q: Assume that the result is true for the case when q = l and then prove that the result is true for q = l + 1. By induction hypothesis the given set Dl = {v1,vi,vn}∪{u3,uk} is the minimal dominating set. We have p = 1, 2, . . . , l− 1, i = 5, 9, . . . , 4l− 3 and k = 7, 11, . . . , 4l− 1 =⇒ Dl = {v1,v5,v9, . . . ,v4l−3,v4l+1}∪{u3, . . . ,u4l−1} is true. To prove that when q = l + 1 the result is true. Let q = l + 1 =⇒ n = 4l + 5 =⇒ Dl+1 = {v1,v5, . . . ,v4l+1}∪{u3, . . . ,u4l+3}. Dl+1 = Dl ∪D1 = {v1,v5,v9, . . . ,v4l−3}∪{u3, . . . ,u4l−1}∪{v1,v4l+1,u4l+3} This proves that the result is true for q = l + 1. =⇒ |Dm| = d n 2 e. Hence γ(CLn) = d n 2 e. Case 3. If n ≡ 2 mod 4 then Dm = {v1,vi}∪{u2,uk} where i = 4p, k = 4p + 2 and p = 1, 2, . . . , n− 2 4 . Let n = 4q + 2 and q ≥ 1. When q = 1 =⇒ n = 6, D1 = {v1,v4,u2,u6} = {v1,vn−2,u2,un}. See Fig.2. Induction Hypothesis on q: Assume that the result is true for the case when q = l and we need to prove that the result is true for q = l + 1. By induction hypothesis the given set Dl = {v1,vi}∪{u2,uk} is the minimal dominating set. Therefore p = 1, 2, . . . , l, i = 4, 8, . . . , 4l and k = 6, 12, . . . , 4l + 2. =⇒ Dl = {v1,v4,v8, . . . ,v4l}∪{u2, . . . ,u4l−2,u4l+2} is true. To prove: The result is true for q = l + 1. When q = l + 1 =⇒ n = 4l + 6 =⇒ Dl+1 = {v1,v4, . . . ,v4l+4}∪{u2, . . . ,u4l+6}. Dl+1 = Dl ∪D1 = {v1,v4,v8, . . . ,v4l}∪{u2, . . . ,u4l−2,u4l+2}∪{v1,v4l+4,u4l+6} This proves that the result is true for q = l + 1. =⇒ |Dm| = n + 2 2 Therefore, it is obtained that γ(CLn) = n + 2 2 . Case 4. If n ≡ 3 mod 4 then Dm = {v1,vi}∪{u3,uk} where i = 4p + 1, k = 4p + 3 and p = 1, 2, . . . , n− 3 4 . Let n = 4q + 3 and q ≥ 1. When q = 1 =⇒ n = 7 =⇒ D1 = {v1,v5,u3,u7} = {v1,vn−2,u3,un}. Induction Hypothesis on q: Assume that the result is true for the case when q = l and we need to prove 34 JEBISHA ESTHER S AND VENINSTINE VIVIK J that the result is true for q = l + 1. By induction hypothesis the given set Dl = {v1,vi}∪{u3,uk} is the minimal dominating set where, n = 4l + 3, p = 1, 2, . . . , l, i = 5, 9, . . . , 4l + 1 and k = 7, 11, . . . , 4l + 3 =⇒ Dl = {v1,v5,v9, . . . ,v4l−3,v4l+1}∪{u3, . . . ,u4l−1,u4l+3} is true. To prove that the result is true when q = l + 1. Let q = l + 1 =⇒ n = 4l + 7 =⇒ Dl+1 = {v1,v5, . . . ,v4l+5}∪{u3, . . . ,u4l+7}. Dl+1 = Dl ∪D1 = {v1,v5,v9, . . . ,v4l−3,v4l+1}∪{u3, . . . ,u4l−1,u4l+3}∪{v1,v4l+5,u3,u4l+7}. This proves that the result is true for q = l + 1. Hence |Dm| = d n 2 e =⇒ γ(CLn) = d n 2 e. � v1 v2 v3 v4v5 v6 u1 u2 u3 u4u5 u6 Figure 1: Prism Graph CL6 v1 v2 v3 v4v5 v6 u1 u2 u3 u4u5 u6 Figure 2: Dominating vertices of CL6 Remark:1 The domination numbers of some prism graphs CLn of all the cases discussed are summa- rized in Table 1. n V E γ n V E γ n V E γ 11 22 33 6 16 32 48 8 21 42 63 11 12 24 36 6 17 34 51 9 22 44 66 12 13 26 39 7 18 36 54 10 23 46 69 12 14 28 42 8 19 38 57 10 24 48 72 12 15 30 45 8 20 40 60 10 25 50 75 13 Table:1 Domination numbers (γ) of prism graphs CLn with vertices V and edges E Theorem 3.2. The domination number of the antiprism graph Qn, where n ≥ 3, is γ(Qn) = d2n5 e. Proof. Let Qn be the antiprism graph with vertex set V and edge set E. The number of vertices in Qn, |V | = 2n given by V = {vj}∪{uj} where j = 1, 2, . . . ,n and the number of edges |E| = 4n. See Fig.3. Case 1. If n ≡ 0 mod 5 then Dm = {v1,vi}∪{u3,uk} where i = 5p + 1, k = 5p + 3 and p = 1, 2, 3, . . . , n−5 5 . The induction method is taken to prove that the above set is one of the minimal dominating set. Let n = 5q and q ≥ 1. When q = 1 =⇒ n = 5 =⇒ D1 = {v1,u3} = {vn−4,un−2}. See Fig.4. Induction Hypothesis on q: Assume that the result is true for the case when q = l and then prove that the result is true when q = l + 1. By induction hypothesis the given set Dl = {v1,vi}∪{u3,uk} is the minimal dominating set. Therefore p = 1, 2, . . . , l− 1, i = 6, 11, . . . , 5l− 4 and k = 8, 13, . . . , 5l− 2. =⇒ Dl = {v1,v6,v11, . . . ,v5l−4}∪{u3, . . . ,u5l−2} is true. MDS FOR PRISM FAMILY OF GRAPH 35 We need to prove that when q = l + 1 the result is true. Let q = l + 1 =⇒ n = 5l + 5 =⇒ Dl+1 = {v1,v6, . . . ,v5l+1}∪{u3, . . . ,u5l+3}. Dl+1 = Dl ∪D1 = {v1,v6, . . . ,v5l−4}∪{u3, . . . ,u5l−2}∪{v5l+1,u5l+3} This prove that the result is true for q = l + 1. =⇒ |Dm| = d 2n 5 e Hence γ(Qn) = d 2n 5 e. Case 2. If n ≡ 1 mod 5 then Dm = {v1,vi,vn−1}∪{u3,uk} where i = 5p+ 1, k = 5p+ 3 and p = 1, 2, 3, . . . , n−6 5 is one of the minimal dominating set. By induction method, let n = 5q+1 and q ≥ 1. When q = 1 =⇒ n = 6, D1 = {v1,v6,u3} = {v1,vn,un−4}. Induction Hypothesis on q: Assume that the result is true for the case q = l and then prove that the result is true for q = l + 1. By induction hypothesis the given set Dl = {v1,vi,vn}∪{u3,uk} is the minimal dominating set. Therefore p = 1, 2, . . . , l− 1, i = 6, 11, . . . , 5l− 4 and k = 8, 13, . . . , 5l− 2. =⇒ Dl = {v1,v6,v11, . . . ,v5l−4,v5l+1}∪{u3, . . . ,u5l−2} is true. To prove: when q = l + 1 the result is true. Let q = l + 1 =⇒ n = 5l + 6 =⇒ Dl+1 = {v1,v6, . . . ,v5l+6}∪{u3, . . . ,u5l+3} Dl+1 = Dl ∪D1 = {v1,v6, . . . ,v5l−4,v5l+1}∪{u3, . . . ,u5l−2}∪{v5l+6,u5l+3} Hence the result is true when q = l + 1. =⇒ |Dm| = d 2n 5 e Hence γ(Qn) = d 2n 5 e. Case 3. If n ≡ 2 mod 5 then Dm = {v1,vi,vn−1}∪{u3,uk} where i = 5p + 1, k = 5p + 3 and p = 1, 2, 3, . . . , n−7 5 . Prove by induction that the above set is one of the minimal dominating sets. Let n = 5q + 2 and q ≥ 1. When q = 1 =⇒ n = 7 and D1 = {v1,v6,u3} = {v1,vn−1,un−4}. Induction Hypothesis on q: Assume that the result is true when q = l and then prove that the result is true when q = l + 1. By induction hypothesis the given set Dl = {v1,vi,vn−1}∪{u3,uk} is the minimal dominating set. So that p = 1, 2, . . . , l− 1, i = 6, 11, . . . , 5l− 4 and k = 8, 13, . . . , 5l− 2. =⇒ Dl = {v1,v6,v11, . . . ,v5l−4,v5l+1}∪{u3, . . . ,u5l−2} is true. To prove that if q = l + 1 then the result is true. Let q = l + 1 =⇒ n = 5l + 7. =⇒ Dl+1 = {v1,v6, . . . ,v5l+6}∪{u3, . . . ,u5l+3} Dl+1 = Dl ∪D1 = {v1,v6,v11, . . . ,v5l−4,v5l+1}∪{u3, . . . ,u5l−2}∪{v5l+6,u5l+3} Hence result is true for q = l + 1. =⇒ |Dm| = d 2n 5 e Thus γ(Qn) = d 2n 5 e. Case 4. If n ≡ 3 mod 5 then Dm = {v1,vi}∪{u3,uk} 36 JEBISHA ESTHER S AND VENINSTINE VIVIK J where i = 5p + 1, k = 5p + 3 and p = 1, 2, 3, . . . , n−3 5 . Prove by induction that the above set is one of the minimal dominating sets. Let n = 5q + 3 and q ≥ 0. Assume q = 0 =⇒ n = 3 =⇒ D1 = {v1,u3} = {vn−2,un} Induction Hypothesis on q: Assume that the result is true for the case when q = l and then prove that the result is true for q = l + 1. By induction hypothesis Dl = {v1,vi}∪{u3,uk} is the minimal dominating set. We have p = 1, 2, . . . , l, i = 6, 11, . . . , 5l − 4, 5l + 1 and k = 8, 13, . . . , 5l − 2, 5l + 3. =⇒ Dl = {v1,v6,v11, . . . ,v5l+1}∪{u3, . . . ,u5l+3} is true. To prove that when q = l + 1 the result is true. Let q = l + 1 =⇒ n = 5l + 8. =⇒ Dl+1 = {v1,v6, . . . ,v5l+6}∪{u3, . . . ,u5l+8} Dl+1 = Dl ∪D1 = {v1,v6, . . . ,v5l+1}∪{u3, . . . ,u5l+3}∪{v5l+6,u5l+8} This implies the result is true when q = l + 1. =⇒ |Dm| = d 2n 5 e Therefore γ(Qn) = d 2n 5 e. Case 5. If n ≡ 4 mod 5 then the minimum dominating set Dm = {v1,vi}∪{u3,uk} where i = 5p + 1, k = 5p + 3 and p = 1, 2, 3, . . . , n−4 5 . Prove by induction that the above set is one of the minimal dominating sets. Let n = 5q + 4 and q ≥ 0. Assume the case q = 0 =⇒ n = 4 =⇒ D1 = {v1,u3} = {vn−3,un−1} Induction Hypothesis on q: Assume that the result is true for q = l and then prove that the result is true when q = l + 1. By induction hypothesis the given set Dl = {v1,vi}∪{u3,uk} is the minimal dominating set. Here p = 1, 2, . . . , l, i = 6, 11, . . . , 5l− 4, 5l + 1 and k = 8, 13, . . . , 5l− 2, 5l + 3 =⇒ Dl = {v1,v6,v11, . . . ,v5l+1}∪{u3, . . . ,u5l+3} is true. We have to prove that when q = l + 1 the result is true. Let q = l + 1 =⇒ n = 5l + 9 =⇒ Dl+1 = {v1,v6, . . . ,v5l+6}∪{u3, . . . ,u5l+8}. Dl+1 = Dl ∪D1 = {v1,v6,v11, . . . ,v5l+1}∪{u3, . . . ,u5l+3}∪{v5l+6,u5l+8} This prove that the result is true when q = l + 1. =⇒ |Dm| = d 2n 5 e Thus γ(Qn) = d 2n 5 e. � MDS FOR PRISM FAMILY OF GRAPH 37 Figure 3: Antiprism Graph Q5 v1 v2 v3 v4 v5 u1 u2 u30u4 u5 v1 v2 v3 v4 v5 u1 u2 u3u4 u5 Figure 4: Dominating vertices of Q5 Remark 2: Table 2 summarises the domination numbers (γ ) of few antiprism graphs Qn with vertices V and edges E. n V E γ n V E γ n V E γ 11 22 44 5 16 32 64 7 21 42 84 9 12 24 48 5 17 34 68 7 22 44 88 9 13 26 52 6 18 36 72 8 23 46 92 10 14 28 56 6 19 38 76 8 24 48 96 10 15 30 60 6 20 40 80 8 25 50 100 10 Table 2: Domination numbers (γ ) of crossed prism graph Qn Theorem 3.3. The domination number of the crossed prism graph Rn, where n ≥ 4 is (1) γ(Rn) = n 2 , if n ≡ 0 mod 4 (2) γ(Rn) = n+2 2 , if n ≡ 2 mod 4. Proof. Let Rn be the crossed prism graph with vertex set V and edge set E. The number of vertices in Rn is |V | = 2n and the number of edges in Rn, |E| = 3n. See Fig.5. For n ≥ 4 the form of minimum dominating set is Case 1. If n ≡ 0 mod 4 then Dm = {v1,vi}∪{u4,uk} where i = 4p + 1, k = 4p + 4 and p = 1, 2, . . . , n− 4 4 . Prove by induction that the above set is one of the minimal dominating sets. Let n = 4q and q ≥ 1. When q = 1 =⇒ n = 4 =⇒ D1 = {v1,u4} = {vn−3,un}. Induction Hypothesis on q: Assume that the result is true when q = l and then prove that the result is true for q = l + 1. By induction hypothesis Dl = {v1,vi}∪{u4,uk} is the minimal dominating set, therefore p = 1, 2, . . . , l− 1, i = 5, 9, . . . , 4l− 3 and k = 4, 8, . . . , 4l. =⇒ Dl = {v1,v5,v9, . . . ,v4l−3}∪{u4, . . . ,u4l−4} is true. We have to prove that when q = l + 1 the result is true. Let q = l + 1 =⇒ n = 4l + 4 =⇒ Dl+1 = {v1,v5, . . . ,v4l+1}∪{u3, . . . ,u4l+4} Dl+1 = Dl ∪D1 = {v1,v5,v9, . . . ,v4l−3}∪{u4, . . . ,u4l}∪{v4l+1,u4l+4} 38 JEBISHA ESTHER S AND VENINSTINE VIVIK J This proves that the result is true for q = l + 1. Hence |Dm| = n 2 =⇒ γ(Rn) = n 2 . Case 2. If n ≡ 2 mod 4 then the one of the minimum dominating set Dm = {v1,vi}∪{u1,uk} where i = 4p, k = 4p + 1 and p = 1, 2, . . . , n− 2 4 . The proof is given by mathematical induction method. Let n = 4q + 2 and q ≥ 1. Assume that q = 1 =⇒ n = 6 and D1 = {v1,v4,u1,u5} = {v1,vn−2,u1,un−1}. See Fig.6. Induction Hypothesis on q: Assume that the result is true for the case, when q = l and then prove that the result is true for q = l + 1. By induction hypothesis Dl = {v1,vi}∪{u4,uk} is the minimal dominating set. Therefore p = 1, 2, . . . , l, i = 4, 8, . . . , 4l and k = 5, 9, . . . , 4l + 1. =⇒ Dl = {v1,v4,v8, . . . ,v4l}∪{u5, . . . ,u4l+1} is true. To prove: When q = l + 1 the result is true. Let q = l + 1 =⇒ n = 4l + 6 =⇒ Dl+1 = {v1,v5, . . . ,v4l+4}∪{u3, . . . ,u4l+5}. Dl+1 = Dl ∪D1 = {v1,v5,v9, . . . ,v4l}∪{u4, . . . ,u4l+1}∪{v4l+4,u4l+5} This proves that the result is true when q = l + 1. =⇒ |Dm| = n + 2 2 . Thus γ(Rn) = n + 2 2 . � v1 v2 v3 v4v5 v6 u1 u2 u3 u4u5 u6 Figure 5: Crossed Prism Graph R6 v1 v2 v3 v4v5 v6 u1 u2 u3 u4u5 u6 Figure 6: Dominating vertices of R6 Remark:3 The domination numbers of various crossed prism graphs Rn for some of the cases n ≡ 0 mod 4 and n ≡ 2 mod 4 are summarized in Table 3. n V E γ n V E γ n V E γ 10 20 30 6 20 40 60 10 30 60 90 16 12 24 36 6 22 44 66 12 32 64 96 16 14 28 42 8 24 48 72 12 34 68 102 18 16 32 48 8 26 52 78 14 36 72 108 18 18 36 54 10 28 56 84 14 38 76 114 20 Table 3: Domination number (γ ) for crossed prism graph Rn with vertices V and edges E MDS FOR PRISM FAMILY OF GRAPH 39 4. Conclusion The idea of computing bounds of domination number for graphs remains to be an active area of research for decades. This led to the focus of investigating the domination number of the family of prism graphs. In this work we have generalized the minimum dominating set for each case of considered graphs and proved using mathematical induction method. 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Email address: jebishaesther@gmail.com Veninstine Vivik J, corresponding author, Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore-641 114, Tamil Nadu, India. Email address: vivikjose@gmail.com 1. Introduction Motivation 2. Preliminaries 3. Computation of domination numbers for the prism graph family 4. Conclusion References