Bounds on the signed domination number of a graph.

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1 Bounds on the signed domination number of a graph. Ruth Haas and Thomas B. Wexler September 7, 00 Abstract Let G = (V, E) be a simple graph on vertex set V and define a function f : V {, }. The function f is a signed dominating function if for every vertex x V, the closed neighborhood of x contains more vertices with function value than with. The signed domination number of G, γ s (G), is the minimum weight of a signed dominating function on G. We give a sharp lower bound on the signed domination number of a general graph with a given minimum and maximum degree, generalizing a number of previously known results. Using similar techniques we give upper and lower bounds for the signed domination number of some simple graph products: the grid P j P k, C j P k and C j C k. For fixed width, these bounds differ by only a constant. Keywords: signed domination, grid graphs, Introduction Let G = (V, E) be a simple graph and v a vertex in V. The closed neighborhood of v, denoted N[v], is the set {u : uv E} {v}. A function f : V {, } is a signed dominating function if for every vertex v V, the closed neighborhood of v contains more vertices with function value than with. We will use the symbol f[v] to denote the sum x N[v] f(x). Thus, f is a signed dominating function if f[v] for all v V. The weight of f, denoted f(g), is the sum of the function value of all vertices in G. The signed domination number of G, γ s (G), is the minimum weight of a signed dominating function on G. This concept was defined in [] and has been studied by several authors including [, 3, 4, 9]. The (standard) domination number of a graph G, γ(g), is similarly defined to be the minimum weight function f : V {0, } such that f[v] for all v V. In section we give a lower bound on the signed domination number of a general simple graph, in terms of V, minimum and maximum degree. Our bound is a modest improvement of the bound of Zhang, Xu, Li, and Liu [9]. The new bound is sharp and implies several known results. Let P j beapathonj vertices, and C j a cycle on j vertices. A j k grid graph in the plane is the cartesian product P j P k. The domination number γ(p j P k ) has been determined for j 6, and has been bounded both above and below for square grids, γ(p j P j ), see [, 6, 8]. In general the exact value of γ(p j P j ) is not known, though there is a linear time algorithm to find a specific value []. In section 3 we give upper and lower bounds for the signed domination number for general finite grids P j P k, C j P k and C j C k. For any particular k, our bounds differ by roughly k. Wealsogive slightly better bounds for the particular case j = 3 and exact results for the case j =. A sharp lower bound on γ s Theorem. For any graph G with minimum degree, maximum degree and V(G) =n, γ s (G) n Department of Mathematics, Smith College, Northampton, MA 0063, USA Department of Mathematics, Amherst College, Amherst, MA 000, USA

2 Proof. Let G be a graph with vertex set V, minimum degree of, and maximum degree of, and let f be a minimum dominating function of G. Partition the vertices of G based on their degree and function value as follows. Let P ={v V f(v) =+ and deg(v) = }; P ={v V f(v) =+ and deg(v) = }; P Θ = {v V f(v) = +and < deg(v) < }; M = {v V f(v) = and deg(v) = }; M ={v V f(v) = and deg(v) = }; M Θ ={v V f(v) = and<deg(v) < }. We also define V = P M, V = P M, V Θ = P Θ M Θ, P = P P P Θ and M = M M M Θ.LetV odd denote the set of all vertices with odd degree. Observe that γ s (G) = f(g) = P M = V M. For any vertex x, f[x], furthermore, if deg(x) is odd, then f[x]. Hence f[x] V + V odd. In this sum f(x) is added a total of deg(x) + times, for each vertex x. Therefore we have f (x)(deg(x) + ) V + V odd. Breaking the sum up into the six summations and replacing f(x) with the corresponding value of or yields x P (deg(x) + ) + x P (deg(x) + ) + x P Θ (deg(x) + ) x M (deg(x) + ) x M (deg(x) + ) x M Θ (deg(x) + ) V + V odd. We know that deg(x) = for all x in P or M,anddeg(x) = for all x in P or M. For any vertex x in either P Θ or M Θ, + deg(x). Therefore, maintaining the inequality results in x P ( + ) + x P ( + ) + x P Θ ( ) x M ( + ) x M ( + ) x M Θ ( + ) V + V odd. Summations can now be removed yielding P ( + ) + P ( + ) + P Θ ( ) M ( + ) M ( + ) M Θ ( + ) V + V odd. For i {,,Θ}, we can replace P i with V i M i. V ( + ) + V ( + ) + V Θ ( ) M ( + ) M ( + ) M Θ ( + + ) V + V odd, V + V +( ) V Θ V odd M + M +( + ) M Θ + M. Adding V V to the LHS and M M and M M to the RHS allows us to pull out V and M terms on each side, respectively. V +( ) V V Θ V odd M + M + M +( ) M +( ) M, V V Θ V odd ( + + ) M +( ) M +( ) P. () We will proceed from here by cases based on the parity of and. Case. and are even. V Θ, V odd, ( ) M,and( ) P are all non-negative, so the inequality is maintained when we drop all these terms from equation, leaving us with V ( + + ) M. Hence M V + +. Since γ s(g) = V M, γ s (G) V. Case. is odd and is even. Since is odd, V odd V. So from equation, V ( + + ) M + V +( ) M +( ) P + V Θ. Since and are of differing parity,. Replacing with and splitting V and V Θ gives V ( + + ) M + M + P + M + P + P Θ + M Θ. Notice we can group the M terms and drop the P terms, giving V ( + + ) M + M. Hence M V Since γ s(g) = V M, γ s (G) V.

3 Case 3. is even and is odd. Since is odd, V odd V. So from equation, V ( + + ) M + V +( ) M +( ) P + V Θ. Again, must be positive, so we get V ( + + ) M + V + M + P + V Θ. Dropping M, and the M and M Θ components of V and V Θ respectively, leaves V ( + + ) M + P =( + + ) M + V M, so ( ) V ( + + ) M. Hence M ( ) V + +. Since γ s(g) = V M, γ s (G) V. Case 4. and are odd. Since both and are odd, V odd V + V. So from equation, V V V Θ V =( ) V ( + + ) M +( ) M +( ) P. The inequality is maintained when we drop non-negative terms from the right hand side, giving ( ) V ( + + ) M. Hence M ( ) V + +. Since γ s(g) = V M, γ s (G) V. If x is a vertex in G of degree 0 or, and f is a signed dominating function, then f(x) =. Hence if, then the lower bound on γ s depends on the number of vertices of degree. The next result shows that the bound given in Theorem is sharp for (G). Theorem. Let and be integers such that. Then there exists a graph G such that γ s (G) = n +, + + where and are respectively the maximum and minimum degree of the vertices in G, and V(G) =n. Proof. Take the union of copies of the complete bipartite graph K ( +), to get a bipartite graph with parts P and M with each vertex in P of degree, and each vertex in M of degree +. The desired graph G will consist of this bipartite graph plus additional edges in the subgraphs induced by P and M respectively as follows. Since P is a set of ( + ) vertices, by the Erdos- Gallai degree sequence characterization, we can construct a regular graph on the vertices of P. Similarly, M is a set of ( ) vertices, so we can construct a ( ) regular graph on the vertices of M. Every ( vertex in P has ) degree, every vertex in M has degree and the total number of vertices is + +. Let f : V {, } be a function that assigns to all vertices in P and to all vertices in M. Any vertex p P will have neighbors in P and neighbors in M. Therefore, since f(p) =, f[p] = or, depending on the parity of. Similarly, any vertex m M will have ( + ) neighbors in P and ( ) neighbors in M. Therefore, since f(m) =, f[m] = or, depending on the parity of. The weight of f is ( +). Soγ s (G) ( +). But by Theorem, γ s (G) ( +). Hence G has a signed domination number equal to the desired bound. Theorem unifies and generalizes some results in the literature. In particular, the following four results are direct consequences of Theorem. As above, n = V(G). Corollary 3. [] For any graph G, ifg is k-regular, γ s (G) n k+. 3

4 Corollary 4. [] For any graph G, ifg is k-regular where k is odd, γ s (G) n k+. Corollary. [7] For any graph G, if (G) 3, γ s (G) n 3. Corollary 6. [7] For any graph G, if (G), γ s (G) 0. These last two are special cases of the next corollary, which bounds the signed domination by maximum degree. This again is a direct consequence of Theorem. Corollary 7. For any graph G with, γ s (G) 4 4+ n and if is odd, γ s(g) 3+ n. 3 Grids In this section we give bounds and some exact formulas for the signed domination of the grids P j P k, C j P k and C j C k for all values of j and k. The notation v i,t will represent the vertex in row i and column t for 0 i j and 0 t k. Theorem 8. The signed domination number for grids satisfies the following bounds. γ s (P P k ) = { k k even k + k odd () (3) For k 3, k k 0(mod 4) γ s (P C k ) = k + k (mod 4) (4) k + k odd γ s (C 3 P k ) = k + () γ s (C 3 C k ) = k (6) 7 k 8 γ s(p 3 P k ) 7 (k mod ) k + (7) 7 k γ s(p 3 C k ) 7 k + 8 (8) In general, when j, k 4, (jk + 4j + 4k 4) γ s(p j P k ) (jk + 8k + 4j) (9) (jk + 4k) γ s(c j P k ) (jk + 4j + 4k + 8) (0) jk γ s(c j C k ) (jk + 4j + 4k 4) () In the proofs that follow we make repeated use of some simple observations. If f is a dominating function, then for any vertex x, f[x] must be at least, and if deg(x) is odd, then f[x] is an even integer, so f[x] must be at least. If f[x] {, }, then we will say that x is efficiently dominated by f. Otherwise, x is inefficiently dominated by f. If deg(x) {, 3}, then there is at most one vertex with function value inn[x]. If deg(x) = 4, then there are at most two vertices with value in N[x]. Let V i denote the set of vertices of degree i {, 3, 4}, andv odd denote the set of vertices of odd degree. For any vertex v V 3, define the neighbor box B[v] of v to be the set containing all vertices whose row and column indices differ from the row and column indices of v by at most, respectively, as shown by the dashed box in Figure. Since vertices in V 3 are on the border of the grid B[v] will contain exactly six vertices for any v V 3. 4

5 u - v w a x y z e b c d Figure : A neighbor box in a section of a grid. Lemma 9. Let G be the grid graph P j P k with 4 j k. Let f be a signed dominating function on G, let V inf be the set of all vertices that are inefficiently dominated by f,andletm and M 3 be the sets of all vertices with value and with degree and 3 respectively. Then V inf M 3 + M 4 Proof. We show that for every vertex v M 3, B[v] contains at least one inefficiently dominated vertex. Consider a vertex v M 3 not adjacent to a corner vertex, as shown in Figure. For f to be a signed dominating function, the vertices u, w, x, y, andz must all have value. If y is to be efficiently dominated, the value of c must be. If x is to be efficiently dominated, the value of both a and b must be. And if z is to be efficiently dominated, the value of both e and d must be. Hence, for there to be no inefficiently dominated vertices in B[v], the values of b, c, andd must all be. But if this is the case, then f[c] <, which contradicts the assumption that f was a signed dominating function. Hence, there must be at least one inefficiently dominated vertex in B[v]. A similar argument shows that if v M 3 is adjacent to a corner vertex, then B[v] contains an inefficiently dominated vertex. If B[u] and B[v] were disjoint for every pair of vertices u and v in M 3, then we would have V inf M 3. If u, v M 3 are on the same side of the grid, then they must be at least distance 3 apart and so B[u] B[v] =. However, if u, v M 3 are on adjacent sides of the grid and both within distance of the corner, then B[u] and B[v] will intersect. If the corner vertex itself is assigned, then no vertex at distance or less can be in M 3. Hence, the total number of inefficiently dominated vertices V inf M 3 (4 M ) = M 3 + M 4. Proof of Theorem 8, equation 9. Let G =P j P k. Suppose f is a minimum signed dominating function for G and define P i ={x V i : f(x) =+}, M i ={x V i : f(x) = } and V inf to be the set of inefficiently dominated vertices under f. f[x] = f(x)( + deg(x)) = 3 P +4 P 3 + P 4 3 M 4 M 3 M 4. Replacing P i by V i M i for i {, 3, 4} and combining like terms gives us f[x] = 3 V +4 V 3 + V 4 6 M 8 M 3 0 M 4. () For any vertex x G, f[x], and if x V odd,thenf[x]. If x V inf,thenf[x] 3, and if x V odd V inf,thenf[x] 4. Therefore we have f[x] V + V odd + V inf. (3) Combining equations and 3 yields 3 V +4 V 3 + V 4 6 M 8 M 3 0 M 4 V + V odd + V inf.

6 columns k- 0 B r o w s. D A E.. j- C Figure : Division of the grid into five regions. By definition, V =jk, V =4, V odd = V 3 =j + k 8 and V 4 =jk j k + 4. Combining these with Lemma 9 yields 4jk 4j 4k M +8 M 3 +0 M 4 +( M + M 3 4). Since M V =4, 8 M, and we get 4jk 4j 4k M. Since γ s (G) = V M, wegetγ s (G) (jk + 4j + 4k 4). This is the desired lower bound. To prove the upper bound we define a signed dominating function g. L et G be the grid P j P k where 4 j k. Divide the grid into five regions as shown in Figure. Explicitly, section A consists of all vertices of degree 4; section B contains all vertices having degree 3 in the 0th (top) row; section C contains all vertices having degree 3 in the (j )st (bottom) row; section D contains all vertices in the 0th (left-most) column, regardless of degree; and section E contains all vertices in the (k )st (right-most) column. Let g(v) = for each vertex in sections B and C. For the vertices in D, which have the form v i,0 for 0 i j, let g(v i,0 ) =, unless i 4(mod), in which case let g(v i,0 ) =. For the vertices in E, which have the form v i,k for 0 i j, let g(v i,k ) =, unless i k + 3(mod), in which case let g(v i,k ) =. Define g on A by repeating the pattern shown in Figure 3(a), beginning in the upper left-hand corner. Circled vertices represent those receiving. A larger example of the pattern for section A is shown in Figure 3(b). Explicitly, for i j, t k, define g(v i,t ) = if (i, t) = (, ) + α(, ) ± β(0, ) ± γ(, 0) or (i, t) = α(, ) ± β(0, ) ± γ(, 0) for nonnegative integer values of α, β and γ. All other vertices get the value +. It is clear by inspection that g[v i,t ] for all 0 i j and0 t k hence g is a signed dominating function. Now we bound the weight of g from above. First, g(b C) = B + C =(k ). By construction at least / of the vertices in A have g(x) = henceg(a) ( 3 ) A = (j )(k ). InE there will be roughly one in vertices assigned depending on the value j mod. In every case, the number of s assigned is at least ( E 4)/. Hence g(e) j j 4 = 3 j + 8. Similarly g(d) 3 j + 8. Combining these formulae gives the desired upper bound: g(g) ( ( ) 3 j (k ) + (k ) + j + 8 ) = ( ) jk + 8k + 4j. Lemma 0. Let f be a signed dominating function on the grid P 3 P k, k 4, with V inf the set of all inefficiently dominated vertices and M 3 the set of all vertices of degree 3 with f(x) =. Then V inf M 3. Proof. Let G be a 3 k grid graph, and let f be a signed dominating function on G. As before, we will show that for every vertex in M 3, except for those in the 0th or (k )st column, there is an inefficiently 6

7 (a) (b) Figure 3: The pattern of - s in section A. dominated vertex in the grid. As in the previous lemma, a vertex v M 3 {v,0,v,k } forces every member of its neighbor box, B[v], to get function value +. There are two cases to consider. If v is a vertex in M 3 {v,0,v,k }, and the other vertex of degree 3 in that column has value +, then the vertex of degree 4 in that column is inefficiently dominated. If the other vertex of degree 3 in the column has value (and hence is also a member of M 3 ), then the vertices of degree 4 in the columns directly to the right and left are both inefficiently dominated. These cases are illustrated in Figure 4. Inefficiently dominated vertices are indicated with arrows. Note that these arguments hold whether or - v - v u (a) - u (b) Figure 4: The forced assignments given f(v) = with (a) f (u) = and (b) f (u) =. not B[v] contains the first or last column of the grid. Proof of Theorem 8, equation 7. For k = 3, we can show that γ s (P 3 P 3 ) = by exhaustion, and the bounds hold. Therefore let us assume that k 4. Let G =P 3 P k. L et f be a minimum signed dominating function on G, and define V i, P i, M i V odd and V inf as before. Similar to the proof of theorem 8, equation 9 we have f[x] = 3 V +4 V 3 + V 4 6 M 8 M 3 0 M 4 (4) and f[x] V + V odd + V inf () Combining these, plugging in the appropriate values for the V x and using the result of Lemma 0 we get: 7

8 Figure : An assignment of s and s to the vertices of a 3 section of G. 3k 6 6 M 8 M 3 0 M 4 k 6 + M 3, 8k 6 M +0 M 3 +0 M 4. Of the two corner vertices v 0,0 and v 0,,atmostonecanhavevalue, for otherwise f[v 0, ]<. Thesameistrueforv k,0 and v k,. Therefore M, and hence 8 4 M. Therefore 8k M +0 M 3 +0 M 4 =0 M. Since γ s (G) = V M, wegetγ s (G) 7k 8. This is the desired lower bound. The upper bound is achieved by the following signed dominating function g. Letgassign and to the vertices by repeating the pattern shown in Figure. In the figure a circled vertex denotes one with g(v) = ; uncircled vertices have g(v) =. Explicitly, g(v i,t ) = for (i, t) = (0, α + ), (, α + 3), (, α + 4) or (, α + ) for nonnegative integer values α. The(k )st (last) column is an exception to this rule in two cases: when the total number of columns k 0modork mod. In each use one fewer in the last column. In the former case, assign g(v i,k ) =+ for i = 0,,. In the later case assign g(v 0,k ) =+, g(v,k ) =, and g(v,k ) =+. The total weight of g is thus g(g) = 7 k + c, where c = k (mod ). Proof of Theorem 8, equation. We exhibit a signed dominating function with this value. Let g be a signed dominating function on P P k, k, which assigns tov i,t when i = 0 and t 0(mod 4) or when i = andt (mod 4), and assigns to all other vertices. To see this function is best possible notice that in any induced C 4 in this graph, at most one vertex can be assigned. The proofs for the other equations of Theorem 8, the cartesian product of a path and a cycle or the product of two cycles, are similar to the proofs given here and are omitted. References [] T. Y. Chang and W. E. Clark, The domination numbers of the n and 6 n grid graphs, J. Graph Theory, 7 (993) [] J. E. Dunbar, S. T. Hedetniemi, M. A. Henning, P. J. Slater, Signed domination in graphs, in: Graph Theory, Combinatorics, and Applications, (John Wiley & Sons, 99) 3-3. [3] O. Favaron, Signed domination in regular graphs, Discrete Math. 8 (996) [4] R. Haas, T. B. Wexler, The signed domination number of a graph and its complement, In preparation. [] E. O. Hare, W. R. Hare, and S. T. Hedetniemi, Algorithms for computing the domination number of k n complete grid graphs. Congr. Numer., :8-9, 986. [6] T. W. Haynes, S. T. Hedetniemi, P. J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 998). 8

9 [7] T. W. Haynes, S. T. Hedetniemi, P, J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker, New York, 998). [8] M. S. Jacobson, L. F. Kinch, On the domination number of products of graphs: I, Ars Combin., 8 (984) [9] Z. Zhang, B. Xu, Y. Li, L. Liu, A note on the lower bounds of signed domination number of a graph. Discrete Math. 9 (999)