Comparative Study of Domination Numbers of Butterfly Graph BF(n)

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1 Comparative Study of Domination Numbers of Butterfly Graph BF(n) Indrani Kelkar 1 and B. Maheswari 2 1. Department of Mathematics, Vignan s Institute of Information Technology, Visakhapatnam , India. indranisarang@yahoo.co.in 2. Department of Applied Mathematics, Sri Padmavati Mahila Visva Vidyalayam, Tirupati , India. maherahul@yahoo.com Abstract Butterfly graphs are very important structures in computer architecture and communication techniques Domination in graphs is an enriched area of research at present. We present results on various domination numbers like total domination, independent domination, accurate domination, efficient domination number etc of BF(n). We also present comparative study of these domination numbers of BF(n). Key Words : Butterfly graph, domination number, total domination number, independent domination number, accurate domination number, efficient domination number. 1. Introduction Domination in a graph, along with its many variations, provides an extremely rich area of study. Berge[5] and Ore[16] were the first to define dominating sets. Among the various applications of domination, the most often discussed is a communication network. A fixed connection communication network is a graph G(V, E) whose nodes represent processors and whose edges represent communication links between the processors. Examples of fixed connection machines are mesh, hypercube, butterfly graphs etc. In this paper we present a comparative study on various domination numbers of butterfly graph BF(n). 2. Butterfly Graph BF(n) Definition : The vertex set V of BF(n) is the set of ordered pairs (α; v) where α {0, 1, 2, n-1} and v = x n-1 x n-2 x 1 x 0 is a binary string of length n where x i = 0 or 1. There is an edge from a vertex (α; v) to a vertex (α ; v ) in V where α α +1 (mod n) and x j = x j j α. A Butterfly graph BF(n) is an n-partite graph with n levels. Each level L k for k = 0, 1,., n-1 1

2 has 2 n vertices and L k = { (k; v) / v = x n-1 x n-2 x 1 x 0, x i = 0 or 1}. Using decimal representation of the binary word we can write L k = { (k; m) / where k = 0, 1, 2,. n-1 and m = x j 2 j, j = 0,1,...n-1) }. 3 Preliminary results For n = 2, 3 the butterfly graphs BF(2) and BF(3) have a special structure. BF(2) is the only butterfly graph having parallel edges and BF(3) is the only butterfly graph having triangles. BF(n) are 4 regular graphs with strong symmetry in structure. From the definition of edges in BF(n) we state the following simple results (without proof). Lemma 1 : Two vertices (k; m) and (k; m ) in the level L k of BF(n) dominate a pair of vertices (k-1; m) and (k-1; m ) in the preceding level L k-1 if m m = 2 k. Lemma 2 : Two vertices (k; m) and (k; m ) in the level L k of BF(n) dominate two vertices (k+1; m) and (k+1; m ) in the succeeding level L k+1 if m m = 2 k+1. Lemma 3 : The minimum number of vertices from level L k needed to dominate all the vertices of levels L k-1 and L k+1 is 2 n-1. 4 Definitions of various domination numbers of a graph. Domination Number: A subset D of vertices of G is called a dominating set of G if every vertex in V \ D is dominated by at least one vertex in D. The minimum number of vertices in a dominating set is called the domination number of G and is denoted by γ(g). Independent domination number: A dominating set D i such that no two vertices in D i are adjacent in G is called an independent dominating set of G. In other words for an independent dominating set the induced subgraph < D i > is a null graph. Cardinality of the minimum independent dominating set is called the independent domination number of G and is denoted by γ i (G). Accurate Domination Number: A dominating set D of G is called an accurate dominating set if the induced subgraph < V \ D > contains a dominating set D 1 of cardinality not equal to cardinality of D. The number of vertices in a minimum accurate dominating set is called the accurate domination number and is denoted by γ a. Efficient Domination Number : A subset P of vertex set V of a graph G, is said to be an efficient dominating set if P is a dominating set and every vertex of V \ P is adjacent to exactly 2

3 one vertex of P. Minimum cardinality of such a set P is called an efficient domination number of G and is denoted by γ p. This parameter is also known as perfect domination in the literature. Inverse Domination Number : Let D be a minimum dominating set of G. If V \ D contains a dominating set D 1 of G then D 1 is called an inverse dominating set with respect to D. The inverse domination number is the number of vertices in a minimum inverse dominating set of G. Total Domination Number: A subset D of vertices of G is called a total dominating set of G if every vertex in V is dominated by at least one vertex in D. The minimum number of vertices in a total dominating set is called the total domination number of G and is denoted by γ t (G). Matching domination Number 5. Main Results : From [ ] paper we state the domination number of BF(n) and present dominating sets for BF(2), BF(3), BF(4) and BF(5). γ(bf(n)) = 2 if n = 2 = 6 if n = 3 = (2k+r) 2 n-1 if n = 4k+r for r = 0,1,2 and k ε Z + = (k+1) 2 n if n= 4k+3 for k ε Z +. Figure 1 : All possible dominating sets of BF(2) Figure 2 : Dominating set of BF(3) Figure 3 : Dominating sets of BF(4) Figure 4 : Dominating set of BF(5) From the above figures we make following observations : 1. Vertices in two colour sets both form dominating sets for BF(n) and hence each of these dominating sets (level wise complement vertex sets) is an inverse dominating set. As cardinality of both sets are equal we get that γ -1 (BF(n)) = γ(bf(n)). 2. For n = 2, 3, the dominating sets are independent sets and hence γ i (BF(n)) = γ(bf(n)) for n = 2, 3. For n = 4, we can make another choice of 16 vertices into dominating set, which forms an independent set. All the vertices in L 3 are dominated by the winged edges from 3

4 L 0 vertices included in D. Similarly we can make such a choice for higher values of n and hence γ i (BF(n)) = γ(bf(n)) for all n. Figure 5 : Independent Dominating sets of BF(4) 3. For n= 2, 4 vertices in V\ D are adjacent to exactly one vertex in D and hence D is an efficient dominating set. But for n 3, 5, it is not true so a minimum dominating set is not efficient and if we add some more vertices to D more adjacencies are added hence efficient dominating set does not exist for n 2, 4, 4k. Hence γ e (BF(n)) = γ(bf(n)) for n = 2, 4, 4k. Figure 13 : Minimal Efficient Dominating set for BF(4) Result 2 : from [ ] we get that accurate domination number of BF(n) is γ a (BF(n)) = γ(bf(n)) +1 for n = 2, 3, 4k = γ(bf(n)) for n 4k. Result 3 : From [ ] we get that the Total Domination number of BF(n) is γ t (BF(n)) = γ(bf(n)) +2 for n= 2 and 3 = γ(bf(n)) for n > 4. Here we present few total dominating sets of BF(n) for n = 2, 3, 4 and then discuss about matching domination number of BF(n). Figure 6 : Total dominating sets of BF(2) Figure 7 : Total dominating sets of BF(3) Figure 8 : Total dominating sets of BF(4) From the above figures we claim that these total dominating sets form a perfect matching in D and as we have seen that the dominating sets of these graphs posses no matching, the total dominating set of BF(n) forms a minimal matching dominating set. By the symmetric and regular structure of BF(n) we claim that there is a minimal total dominating set which is a minimal matching dominating set and hence γ t (BF(n)) = γ m (BF(n)) Conclusion : 4

5 Domination Number n = 2 n = 3 n = 4 n = 4k n = 4k+1 n = 4k+2 n = 4k+3 Domination γ(bf(n)) k 2 n (2k+1) 2 n-1 (k+1) 2 n (k+1) 2 n Non-split γ ns (BF(n)) k 2 n (2k+1) 2 n-1 (k+1) 2 n (k+1) 2 n Independent γ i (BF(n)) k 2 n (2k+1) 2 n-1 (k+1) 2 n (k+1) 2 n Inverse γ -1 (BF(n)) k 2 n (2k+1) 2 n-1 (k+1) 2 n (k+1) 2 n Accurate γ a (BF(n)) k2 n + 1 (2k+1) 2 n-1 (k+1) 2 n (k+1) 2 n Efficient γ e (BF(n)) k 2 n Total γ t (BF(n)) k 2 n (2k+1) 2 n-1 (k+1) 2 n (k+1) 2 n Matching γ m (BF(n)) k 2 n (2k+1) 2 n-1 (k+1) 2 n (k+1) 2 n From above table we observe following relations between various domination numbers of BF(n). 1. γ t (BF(n)) = γ m (BF(n)) for all n 2. γ a (BF(n)) = γ(bf(n)) +1 for n = 2, 3, 4k and γ a (BF(n)) = γ(bf(n)) for n 4k. 3. γ(bf(n)) = γ -1 (BF(n)) = γ ns (BF(n)) = γ i (BF(n)) for all n 4. γ(bf(n)) = γ e (BF(n)) for n = 2, 4k References : [1] Allan, R.B. Laskar, R., On Domination and Independent Domination Number of a Graph, Discrete Maths 23, (1978), pp [2] Allan, R.B. Laskar, R Hedetniemi, S.T., A note on Total Domination, Discrete Maths 49, (1984), pp [3] Barth, D. Raspaud, A., Two edge-disjoint Hamiltonian cycles in the Butterfly Graph, Information Processing Letters, 51, (1994), [4] Berge, C., Theory of Graphs and Its Applications, Methuen, London (1962). [5] Berge, C., Graphs and Hypergraphs, North Holland Amsterdam (1973). [6] [7] Cockayne, E J., Dawes, R.M. Hedetniemi, S.T., Total Domination in Graphs, Networks, Vol. 10, (1980),

6 [8] Cockayne, E.J. Hartnell, B.L. Hedetniemi, S.T. Laskar, R., Efficient Domination in Graphs, (1988), Clemson Univ., Dept of Mathematical Science, Tech. Report, 558. [9] Chen, G. Fransis, C.M., Comments on A New Family of Cayley Graph Interconnection networks of Constant Degree 4, IEEE Transactions on parallel and Distributed Systems, Vol. 8, 12, (1987), [10] Haynes, T.W. Hedetniemi, S.T. Slater, P.J., Fundamentals of Domination in graphs, Marcel Dekker Inc. New York, (1998). [11] Kattimani, M.B., Some New Contributions to Graph Theory, PhD Thesis, Gulberga Univ, India, (2000). [12] Kelkar Indrani, B Maheswari, Accurate Domination Number of Butterfly Graphs, Chamchuri Journal of Mathematics, Vol 1 (2009) Number 1, pp [13] Kelkar Indrani, B Maheswari, Domination Number of Butterfly Graph, Journal of Pure and Applied Physics, July-Aug [14] Kelkar Indrani, B Maheswari, Total Domination Number of Butterfly Graph, communicated. [15] Laskar, R. Walikar, H.B., On Domination Related Concepts in Graph Theory, Lecture Notes in Maths 885, Springer-Verlang, (1980), [16] Ore, O., Theory of Graphs, Amer, Maths. Soc. Colloq. Pub. 38, Providence (1962). 6