On Signed Domination of Grid Graph

Let G(V, E) be a finite connected simple graph with vertex set V(G). A function is a signed dominating function f : V(G)→{−1,1} if for every vertex v ∈ V(G), the sum of closed neighborhood weights of v is greater or equal to 1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate the signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 6, 7 and arbitrary n.


Introduction
Let G be a finite simple connected graph with vertex set V(G). The neighborhood of v, denoted N(v), is set {u: uv ∈ E(G)} and the closed neighborhood of v, denoted N [v], is set N(v) ∪ {v}. The function f is a signed dominating function if for every vertex v ∈ V, the closed neighborhood of v contains more vertices with function value 1 than with −1. The weight of f is the sum of the values of f at every vertex of G. The signed domination number of G, γ s (G), is the minimum weight of a signed dominating function on G.
In [1] [2] [3] [4], Dunbar et al. introduced this concept, in [5] Haas and Wexler had found the signed domination number of P 2 × P n and P 2 × C n . In [6] Hosseini gave a lower and upper bound for the signed domination number for any graph. In [7] Hassan, Al Hassan and Mostafa had found the signed domination number of P m × P n for m = 3, 4, 5 and arbitrary n.
We consider when we represent the P m × P n graph. The weight of the black circle is 1, and the white circles refer to the graph vertices which weight −1.

Main Results
In this paper we will show tow theorem to find the signed domination number of Cartesian product of P m × P n .  Let ƒ be a signed dominating function of (P 6 × P n ), then for any j were 2 ≤ j ≤ n − 3, then . We discuss the following cases: Case a. |B j | = 4: we notice that the first and last columns can't include four of the B set vertices, but in the case 2 ≤ j ≤ n − 3 and |B j | = 4, then the vertices (1, j), (3, j), (4, j), (6, j) ∈ B, and all of the j − 1 th , j + 1 th column's vertices don't contain any one of the B set vertices, so the (1, j + 2), (6, j + 2) vertices, then the j + 2 th column includes three of the B set vertices at most (Figure 1).

Case b. |B j | = 3:
We discuss the following cases: b-1. If (1, j), (3, j), (4, j) ∈ B then both of the j − 1 th , j + 1 th columns include at most one of the B set vertices, then the j + 2 th column includes at most three of the B set vertices. b-2. If (1, j), (3, j), (5, j) ∈ B then the j − 1 th and j + 1 th columns include at most two of the B set vertices, and the j + 1 th column includes three of the B set vertices. b-3. If (1, j), (3, j), (6, j) ∈ B then both of the j − 1 th , j + 1 th columns include at most one of the B set vertices. And the j + 2 th column includes two of the B set vertices.
b-4. If (1, j), (4, j), (5, j) ∈ B then only one of the j − 1 th , j + 1 th columns include at most one of the B set vertices, so (1, j + 2) ∈ A, then the j + 2 th column includes at most three of the B set vertices. b-5. If (1, j), (4, j), (6, j) ∈ B then both of the j − 1 th , j + 1 th columns include at most one of the B set vertices. Also (1, j + 2), (4, j + 2) and (6, j + 2) ∈ A then only two of the j + 2 th vertices belong to B set. b-6. If (2, j), (3, j), (6, j) ∈ B then only one of the j − 1 th , j + 1 th column's vertices belong to the B set vertices, then the j + 2 th column include at most four of the B set vertices (Figure 2). Case c. |B j | = 2: We discuss the following cases: c-1. If (1, j), (3, j) ∈ B then all of the j − 1 th , j + 1 th , j + 2 th columns include at most two of the B set vertices (Figure 3). c-2. If (1, j), (4, j) ∈ B and the j − 1 th column include two of the B set vertices then the j + 1 th column include at most one of the B set vertices, so the j + 2 th column include at most three vertices (Figure 4). c-3. If (1, j), (5, j) ∈ B or (1, j), (6, j) ∈ B, then all of the j − 1 th , j + 1 th , j + 2 th columns include at most two of the B set vertices ( Figure 5). c-4. If (2, j), (3, j) ∈ B then if the j − 1 th column includes two of the B set vertices, then the j + 1 th column includes at most one of the B set vertices, so the j + 2 th column includes at most three vertices ( Figure 6).   c-5. If (2, j), (4, j) ∈ B then the j − 1 th column includes at most three of the B set vertices, it is (2, j − 1), (4, j − 1), (6, j − 1) ∈ B, so the j + 1 th column includes one of the B set vertices, also the j + 2 th column includes three of the B set vertices and both of the j − 2 th , j + 3 th columns don't include any one of the B set vertices, so the j + 4 th column includes four of the B set vertices and the j − 3 th column includes three of the B set vertices. then the eight columns include sixteen of the B set vertices. In other cases stay c-6. If (2, j), (5, j) ∈ B then all of the j − 1 th , j + 1 th , j + 2 th columns include at most two of the B set vertices (Figure 8).     (3, j), (4, j) ∈ B then all of the j − 1 th , j + 1 th , j + 2 th columns include at most two of the B set vertices (Figure 9). Case d. |B j | = 1: We discuss the following cases: d-1. If (1, j) ∈ B or (3, j) ∈ B or (4, j) ∈ B or (6, j) ∈ B then the j − 1 th column includes at most three of the B set vertices also both of the j + 1 th , j + 2 th columns include at most two of the B set vertices ( Figure 10). d-2. If (2, j) ∈ B or (5, j) ∈ B then both of the j − 1 th , j + 1 th columns includes at most three of the B set vertices, and the j + 2 th column includes at most one of the B set vertices ( Figure 11).
From the previous cases we conclude ( ) To find the upper bound of the signed domination number of (P 6 × P n ) graph, let's define ( Figure 12).  Figure 9. Case c-7.  Case n ≡ 1 (mod 5). If B is the previously defined set and represents the vertices have the weight −1, then every one of the P 6 × P n vertices achieves the signed dominating function, and |B| ≥ 2n, then: Case n ≡ 2 (mod 5).
In this case, we delete one of the two vertices (3, n) or (4, n) from the previously defined set B vertices, then the signed domination number will increase by 2 than the signed domination number in case of n ≡ 1 (mod 5), and ƒ remains a signed dominating function of the graph. Consequently: Let f be a signed domination function of (P 7 × P n ), and B the graph vertices set which having the weight −1, Then for any j were 1 ≤ j ≤ n − 1, then Except the following cases: (3, j), (5, j) ∈ B, (1, j), (3, j), (5, j) ∈ B, (2, j), (3, j), (5, j) ∈ B or (3, j), (5, j), (7, j) ∈ B. Then For any j were 1 ≤ j ≤ n then |B j | ≤ 4.
Case c. |B j | = 2: c-1. If (1, j), (4, j) ∈ B or (1, j), (7, j) ∈ B then both of the j − 1 th , j + 1 th columns include at most two of the B set vertices, then the j − 1 th , j th , j + 1 th columns Open Journal of Discrete Mathematics      include at most six of the B set vertices, as any two columns include at most six vertices (Figure 29).
c-2-1. If (3, j − 1) ∈ B the j + 1 th column includes at most three of the B set vertices, in this case the j + 2 th column includes at most one of the B set vertices, and the j + 3 th column includes at most three vertices. Or the j + 2 th column includes two of the B set vertices and the j + 3 th column includes at most three vertices.
c-2-2. If (4, j − 1) ∈ B then the j + 1 th column includes at most three of the B set vertices, in this case (3, j + 1), (5, j + 1), (6, j + 1) ∈ B and (2, j + 2) ∈ B, so (2, j + 3), (4, j + 3), (5, j + 3), (7, j + 3) ∈ B, then the j − 2 th column includes at most one of the B set vertices, then . And according to lemma 2-1 note |B j+5 | + |B j+6 | ≤ 6, so |B j+7 | ≤ 6. Then every ten successive columns include at most twenty four of the B set vertices (Figure 30). c-3. If (1, j), (5, j) ∈ B then the j − 1 th column includes at most three of the B set vertices, so the j + 1 th and j + 2 th columns includes at most two of the B set vertices, and the j + 3 th column includes at most three vertices (Figure 31). c-4. If (1, j), (6, j) ∈ B then the j − 1 th column includes at most three vertices, in this case the j + 1 th column includes at most two of the B set vertices, also the j + 2 th column includes three of the B set vertices, and the j + 3 th column includes at most two vertices (Figure 32). c-5. If (2, j), (3, j) ∈ B then the j − 1 th column includes at most three of the B set vertices, then the j + 1 th column includes two of the B set vertices which are (5, j + 1), (6, j + 1), also (1, j + 2), (3, j + 2), (4, j + 2)∈ B, and the j + 3 th column includes only one of the B set vertices (Figure 33). c-6. If (2, j), (4, j) ∈ B then the j − 1 th column includes at most three of the B set vertices, so the j + 1 th column includes at most two of the B set vertices, in this case the j + 2 th column includes at most three of the B set vertices, and the j + 3 th column includes at most two vertices (Figure 34). c-7. If (2, j), (5, j) ∈ B then the j − 1 th column includes at most three of the B set vertices, and the j + 1 th column includes two of the B set vertices, then the j + 2 th , j + 3 th columns include at most five of the B set vertices (Figure 35). Open Journal of Discrete Mathematics     c-8. If (2, j), (6, j) ∈ B then both of the j − 1 th , j + 1 th columns include at most three of the B set vertices, so the j + 2 th column includes at most one of the B set vertices, and the j + 3 th column includes at most three vertices (Figure 36). Open Journal of Discrete Mathematics   c-9. If (3, j), (4, j) ∈ B then the j − 1 th column includes at most three of the B set vertices, then the j + 1 th column includes at most three of the B set vertices, then the j + 2 th column includes only one of the B set vertices, and the j + 3 th column includes at most three vertices (Figure 37). c-10. If (3, j), (5, j) ∈ B then the j − 1 th column includes at most four of the B set vertices, so the j + 1 th column includes at most two vertices, then the j + 2 th column includes at most three of the B set vertices, and the j + 3 th column includes at most one vertex (Figure 38). Case d. |B j | = 1: In this case the j + 1 th , j + 2 th columns include at most five of the B set vertices, so if the j + 3 th , j + 4 th columns include six of the B set vertices, then the number of the vertices in the five columns is less or equal to 12 (Figure 39).
We note from all the previous cases       Case n ≡ 0, 2 (mod 5). According to lemma 2-2, then in case n ≡ 0 (mod 5), we delete the vertices (3, n), (6, n), so in case n ≡ 2 (mod 5), we delete the vertex (4, n). Then the signed domination number will increase by 4. Case n ≡ 1, 3 (mod 5). When we add one column on case n ≡ 0 (mod 5), note that the number of vertices will increase by 7, and the number of set B vertices will increase by 2, in this When we add three columns on case n ≡ 0 (mod 5), note that the number of vertices will increase by 21, and the number of set B vertices will increase by 5, in

Conclusion
In this paper, we studied the signed domination numbers of the Cartesian product of two paths P m and P n for m = 6, 7 and arbitrary n. We will work to find the signed domination numbers of the Cartesian product of two paths P m and P n for arbitraries m and n, and special graphs.