Lower Bounds for the Exponential Domination Number of C m C n

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1 Lower Bounds for the Exponential Domination Number of C m C n Chassidy Bozeman Joshua Carlson Michael Dairyko Derek Young Michael Young, April 6, 06 Abstract A vertex v in an exponential dominating set assigns weight ( ) dist(v,u) to vertex u. An exponential dominating set of a graph G is a subset of V (G) such that every vertex in V (G) has been assigned a sum weight of at least. In this paper the exponential dominating number for the graph C m C n, denoted by γ e (C m C n ) is discussed. Anderson et. al. [] proved that γ e(c m C n ) 3 and conjectured that 3 is also the asymptotic lower bound. We use a linear programing approach to sharpen the lower bound to ɛ(m,n). Keywords. exponential domination, domination, grid, linear programming, mixed integer programming Introduction Given a graph G, a weight function of G is a function w : V (G) V (G) R. For u, v V (G), we say that u assigns weight w(u, v) to v. For a set S V (G) we denote w(s, v) := s S w(s, v). Department of Mathematics, Iowa State University, Ames, IA 500, USA. {cbozeman, jmsdg7, mdairyko, ddyoung, myoung}@iastate.edu Corresponding author.

2 Similarly, w(v, S) := s S w(v, s). For two weight functions of G, w and w, we say w w, if w (u, v) w(u, v) for all u, v V (G). Let D V (G) and w be a weight function of G. The pair (D, w) dominates the graph G, if for all v V (G), w(d, v). The standard definition of domination is where w(u, v) is if v is in the closed neighborhood of u, and 0 if it is not. This type of domination has been widely studied (see [4], [7]). Another well-studied type of domination is total domination, in which w(u, v) is if v is in the neighborhood of u and 0 if it is not (see [5], [8]). There is also k-domination and k-distance domination (see [], [6]). In k-domination w(u, v) is if v is in the neighborhood of u, if k u = v, and 0 otherwise. In k-distance domination w(u, v) is if the distance from u to v is at most k, and 0 otherwise. For exponential domination, w(u, v) = ( )dist(u,v), where dist(u, v) represents the length of the shortest path from u to v. An exponential dominating set of a graph G is a set D V (G) such that (D, w) dominates G when w(u, v) = ( )dist(u,v). The exponential domination number of G, denoted by γ e (G), is the cardinality of a minimum exponential dominating set. This type of exponential domination has also been referred to as porous exponential domination. Some work has been done in non-porous exponential domination, where dist(u, v) represents the length of the shortest path from u to v that does not have any internal vertices that are in the dominating set (see [3]). It is described in [3] that applications of exponential domination relate to the passing of information, and specifically models how information can be spread from a speaker through a crowd. Thus, the exponential domination number represents the minimum number of speakers needed to successfully convey a message to every individual within a crowd. Within exponential domination, there has been research on the exponential domination number of C m C n, where is the Cartesian product. Anderson et. al. [], found lower and upper bounds for γ e (C m C n ). The following theorem shows a sharp upper bound for the exponential domination number of the graph C 3m C 3n.

3 Theorem. [] For all m and n, γ e (C 3m C 3n ) (3m)(3n) 3. The proof of Theorem is constructive. An exponential dominating set is created by choosing a vertex from each row and colu of a 3 3 grid and then periodically tiling C 3m C 3n with the grid and selecting all the corresponding vertices. This argument was extended to C m C n, for m, n arbitrarily large. Theorem. [] γ e (C m C n ) lim m,n 3. A lower bound can be attained in the following way: Observe that when m and n are large enough, for each v V (C m C n ) there exists 4i vertices u V (C m C n ) such that dist(v, u) = i, when i is a positive integer. So w(v, V (C m C n )) + 4i i = 8. This implies i= 8 γ e(g). However this bound can be improved to by adjusting the weight function so that v assigns weight to itself, resulting in w(v, V (C m C n )) 7. 7 A better lower bound was attained in [] by showing that the weight function could be adjusted so that each vertex assigns.5 less than in the original weight function. Theorem 3. [] For all m, n 3, < γ e(g). The above theorems led to the following conjecture. Conjecture. [] For all m and n, 3 γ e(c m C n ). 3

4 In this paper, a better lower bound for γe(cm Cn) is produced. In section, we use linear programming to minimize how much weight is necessary for each vertex in an exponential dominating set to assign. This leads to improved lower bounds in section 3. For the remainder of the paper, we refer to w as the weight function w(u, v) := Linear Program ( ) dist(u,v). Let D = {d, d,..., d } be an exponential dominating set of C m C n. Given an odd positive integer r, let G v be the subgraph of C m C n that is an r r grid centered vertex v V (C m C n ), with V (G v ) = {v, v,..., v r }. Let I v be the set of interior vertices of G v. In this section, a linear program is created where the sum of the weights assigned to the vertices in I v is minimized. The minimum value attained is of the form I v + k, where 0 < k. A new weight function is then created, which still dominates with D and has v assigning k less weight than before. A sequence of weight functions will be constructed recursively.. The Grid First, we naturally partition V (C m C n ). For each v i V (G v ), define S i to be the set of vertices w V (C m C n ) such that the distance between v i and w is less than the distance between w and any other vertex in G v. Notice that S i = {v i }, if v i I v. For i r, let x i = w(s i, v i ). Therefore, if ( i, j r, then w(s i, v j ) = x ) dist(vi,v j ) i. We define Γ = V (G) \ r i= S i and for j r, let ɛ j = w(γ, v j ). Thus, w(d, v j ) r i= w(s i, v j ) + ɛ j = r i= x i ( ) dist(vi,v j ) + ɛ j. Observe that V (Γ) m + n and dist(γ, V (G v )) as m, n. Thus 0 ɛ j (m + n ) ( ) dist(γ,v (Gv)) for each j r. Therefore 4

5 assuming that ɛ = ɛ r j= (m + n ) r j= ɛ j, ( ) dist(γ,v (Gv)) which means ɛ 0 as m, n. r (m + n ) ( ) dist(γ,v (Gv)), Example. Consider the graph G = C 6 C 8 as shown in Figure. For the simplicity of the figure, we remove the edges of G. Choose r = 3 and construct G v5 with V (G v5 ) = {v, v,..., v 9 }. We then label the corresponding sets S, S,..., S 9, Γ. For instance, observe that S 3 consists of all vertices in G whose distance to v 3 is smaller than their distance to any other vertex of G v5. Figure : C 6 C 8 with edges removed. The Program Lemma below proves how to get a lower bound for the exponential domination number of a graph C m C n, given that each vertex in the dominating set assigns more weight to V (C m C n ) than needed. 5

6 Lemma. Let G be a graph and D = {d, d,..., d } be an exponential dominating set of G and ρ R such that w(d j, V (G)) ρ for all j. If there exists a sequence of weight functions {w j } j=0, where w = w 0, and the following conditions are satisfied for j, ) w j < w j, ) f (D, w j ) dominates G, and 3) there exist k R such that 0 < k w j (d j, V (G)) w j (d j, V (G)), then ρ k < V (G). Proof. Let {w j } j=0 be such a sequence of weight functions for the exponential dominating set D. Conditions ) and 3) imply that k < w 0 (d j, V (G)) w (d j, V (G)) for all d j D. Therefore, k < [w 0 (d j, V (G)) w (d j, V (G))]. j= Since condition ) gives w (D, v) for all v V (G), then Combining these inequalities gives, V (G) w (d j, V (G)). j= k + V (G) < w 0 (d j, V (G)) j= ρ = ρ. j= This implies that ρ k < V (G). 6

7 We now construct a recursive set of weight functions that satisfy the conditions of Lemma for some k. Let d j D and w j be a weight function such that (D, w j ) dominates C m C n. Let G d be the r r grid G dj and I = I dj. Recall that x i = w(s i, v i ). Let A be the r r matrix such that [A] ij = ( ) dist(vi,v j ). Let x = [x, x,..., x r ] T and w = [w(d, v ), w(d, v ),..., w(d, v r )] T. Thus, w A x. In fact, if w 0 < w, then w 0 < Ax. Let c be the real-valued vector such that c T x = v i I w j (D, v i ). The objective function in the linear program will be c T x, where x is a vector of r variables. Since (D, w j ) dominates C m C n, w j, where is the all s vector. Therefore, A x; hence, Ax is a constraint. Let b be the real-valued vector whose ith entry is + ( dist(vi,d j ) ) + ɛj if v i I and 8 otherwise. The constraint Ax b will be added to ensure that for each vertex in I the weight assigned from d j can be decreased by the appropriate amount. Consider the following linear program: min c T x s.t. Ax Ax b x 0. Define x to be an optimal solution to the linear program and x min to be the value attained. Obviously, I + ɛ < x min, so 0 < k = x min ɛ I. For each i with v i I, let r ) dist(vi,v s) ( y i = x i s= Thus, 0 y i ( ) dist(vi,d j ) and y i = k. v i I ɛ i. Remark. Note that the weights function {w j } j=0 satisfy conditions ), ), and 3) of Lemma. Clearly w j < w j, so ) is satisfied. For each v V (C m C n ) \ I, w j (D, v) = w j (D, v). For each v i I, w j (D, v) = w j (D, v) k = + ɛ i. This implies (D, w j ) dominates C m C n so ) is satisfied. Lastly, w j (d j, V (C m C n )) = w j (d j, V (C m C n )) k, so 3) is satisfied. 7

8 3 Main Results In this section, we use Lemma and the weight functions {w j } j=0 constructed in Section to attain a lower bound for the exponential domination number of C m C n. Theorem 4. For all m, n 3, ɛ γ e(c m C n ), where ɛ 0 as m, n. Moreover, ɛ = 0 when m and n are both odd. Proof. Let D be a minimum exponential dominating set. For each v D, let G v be the 3 3 grid centered at v. Observe that here we have that ρ = 8 and that w(v, V (G)) 8 for all v D. The solution to the corresponding linear program is x min = Therefore, it follows that k = ɛ = ɛ, so γ ɛ e(c m C n ) by Lemma. The linear program created in Section can be constructed in the form of a mixed integer linear program by adding the constraints x i = 0 or, when v i I v. Then the attained k is ɛ v by choosing a 9 9 grid as G v. However, the weight function can only be adjusted at a vertex v D, such that no vertices in D I v have been adjusted. Rather than using the linear program for all the vertices in D, we will use it for the vertices in D that are relatively close together and use the mixed integer linear program for those vertices in D that are not close to the other vertices of D. Theorem 5. Let D be an exponential dominating set of C m C n and α be the number of vertices in D that are not within a 7 7 grid of any other vertex in D. Then where ɛ 0 as m, n α ɛ γ(c m C n ), Proof. Let D be the set of vertices that are not within a 7 7 grid of any other vertex in D; so D = α. Choose r = 9 in G v and let b be the real-valued vector whose ith entry is 0 if v i I v and 4 otherwise. By taking geometric sums, it is easy to see that x i 4, for all i. 8

9 The linear program min c T x s.t. Ax Ax b x b x 0. will attain a minimum of So each vertex in D can be adjusted by ɛ 49 = 7.06 ɛ to ɛ, for some ɛ 0. As before, the vertices of D \ D can be adjusted to ɛ, for some ɛ 0. So ( α)( ɛ ) + α( ɛ ), which implies ( α + ɛ). Corollary is a direct result of combining Theorems and 5. Corollary. Let D be an exponential dominating set of C m C n. For m and n large enough, the number of vertices in D that are not within a 7 7 grid of any other vertex in D is at most.7. References [] M. Anderson, R. Brigham, J. Carrington, R. Vitray, and J. Yellen, On exponential domination of C m C n, AKCE Int. J. Graphs Comb 6 (009), [] M. Chellali, O. Favaron, A. Hansberg, and L. Volkmann, k-domination and k-independence in graphs: a survey, Graphs and Comb. 8 () (0), 55. [3] P. Dankelmann, D. Day, D. Erwin, S. Mukwembi, and H. Swart, Domination with exponential decay, Discrete Math. 309 (9) (009), [4] D. Gonçlaves, A. Pinlou, M. Rao, and A. Thomassé, The domination number of grids, SIAM J. Discrete Math., 5 (3), (0), [5] S. Gravier, Total domination number of grid graphs, Discrete Appl. Math., (-3) (00), 9 8. [6] A. Hansberg, D. Meierling, and L. Volkmann, Distance domination and distance Irredundance in graphs, Elec. J. Comb. 4 (007), #R35. 9

10 [7] T. Haynes, S. Hedetniemi, and P. Slater, Domination in Graphs, Chapman & Hall/CRC Pure and Applied Mathematics, (998). [8] M. Henning, A survey of selected recent results on total domination in graphs, Discrete Math. 309 (), (009),