Computing the zero forcing number for generalized Petersen graphs

: Let G be a simple undirected graph with each vertex colored either white or black, u be a black vertex of G, and exactly one neighbor v of u be white. Then change the color of v to black. When this rule is applied, we say u forces v, and write u → v . A zero forcing set of a graph G is a subset Z of vertices such that if initially the vertices in Z are colored black and remaining vertices are colored white, the entire graph G may be colored black by repeatedly applying the color-change rule. The zero forcing number of G , denoted Z ( G ) , is the minimum size of a zero forcing set. In this paper, we investigate the zero forcing number for the generalized Petersen graphs (It is denoted by P ( n, k ) ). We obtain upper and lower bounds for the zero forcing number for P ( n, k ) . We show that Z ( P ( n, 2)) = 6 for n ≥ 10 , Z ( P ( n, 3)) = 8 for n ≥ 12 and Z ( P (2 k + 1 , k )) = 6 for k ≥ 5 .


Introduction
Let G = (V, E) be a simple undirected graph.Each vertex is colored either white or black.In such a case we say that G has a coloring and the set of all black vertices is called an initial coloring of G.The color-change rule is defined as follows: if u is a black vertex of G and exactly one neighbor v of u is white, then the color of v changes to black.
Given a coloring of G, let A be the set of all black vertices of G.The derived coloring of A, denoted der(A), is the result of applying the color-change rule until no more changes are possible.The zero forcing set for a graph G (ZF S) is an initial coloring Z of G such that der(Z) = G.The zero f orcing number Z(G) is the minimum size of all zero forcing sets of G.The concept of zero forcing set indicates one model of propagation in general networks.It was introduced in [4].the associated terminology has been extended in [5,7,11,12].For example according to [4] if G is a path, an endpoint of G is the zero forcing set for G.If G is a cycle, each set of two adjacent vertices is a zero forcing set.
A contraction of a graph G is the graph obtained by identifying two adjacent vertices of G, and ignoring any loops or multiple edges occurred.A minor of G is a graph obtained by applying a sequence of deletions of edges, deletions of isolated vertices, and contraction of edges.A graph parameter ζ is called minor monotone if for any minor ).The generalized Petersen graph P (n, k) is defined to be the graph with the vertex set and edge set respectively as follows Here, the subscripts are assumed as integers modulo n such that n ≥ 5. Note that, P (n, k) ∼ = P (n, n − k).So, we assume n ≥ 2k + 1. P (n, k) is a 3-regular graph with 2n vertices.The generalized Petersen graph has been studied from several points of view, such as: hamiltonicity [1,3,15], crossing numbers [13,14], spectrum [10] and vertex domination [9].
In Section 3, we show that K k,[ n k ] is a minor of P (n, k) (where [x] is the maximum integer not greater than x).Using this, we conclude that: The graph parameter µ is introduced by Colin de Verdiere in 1990 [6].It is equal to the maximum nullity among all matrices satisfying several conditions.This conditions are stated in Section 3. It is the first parameter of Colin de Verdiere type parameters.There exist a relation between this parameter and the zero forcing number that we apply it for achieving the upper bound.
There exists a comparison between the zero forcing sets and dynamic monopolies in the last section.Note that, in all figures of this paper the vertex

Upper bounds and equalities for Z(P (n, k))
In the following theorem, we obtain an upper bound for Z(P (n, k)), where k n.
The vertex v k+1 is the only white neighbor of the black vertex v 1 and the vertex v 1 forces it.The vertex v 1+(r−1)k is the only white neighbor of the black vertex v 1+(r−2)k and the vertex v 1+(r−2)k forces it.
The vertex u n is the only white neighbor of the black vertex u 1 and is forced by it.is the only white neighbor of the black vertex u i(k−1)+s+3 and is forced by it.So, all vertives of graph became black.
In the next theorem we obtain an upper bound for Z(P (n, k)).This bound does not depend on n.
Proof.Let A = {u 1 , u 2 , . . ., u 2k+2 } be an initial coloring of P (n, k) (see Figure 1).The vertex v j is the only white neighbor of the black vertex u j for 2 ≤ j ≤ 2k + 1.It is forced by u j .So, v j ∈ der(A).Now the vertex v 1 is the only white neighbor of the vertex u 1 and is forced by it.Therefore v 2k+2 is the only white neighbor of the black vertex v k+2 .Hence, the vertices v 2 , v 3 ,. . .,v 2k+2 are in der(A).We continue by induction.Let m ≥ 2k + 2 and the color of vertices u m , . . ., u 2 , u 1 v m , . . ., v 2 have been changed to black.It suffices to show that the color of the vertices u m+1 and v m+1 change to black.Note that m ≥ 2k + 2 hence m ≥ m + 1 − 2k ≥ 3. Therefore, the vertex v m+1 is the only white neighbor of the black vertex v m+1−k and u m+1 is the only white neighbor of the black vertex u m .
Proof.By Theorem 2.2, we have Z(P (n, 2)) ≤ 6. Hence it suffices to show that no initial coloring of the graph with five vertices can be a zero forcing set.Let A be such an initial coloring.By checking all of possible cases we show that |der(A)| ≤ 10 < 2n = |P (n, 2)|.We have illustrated all cases, unless the trivial or similar ones, in the following figures.In each figure the white vertices are the vertices that will change to black by A. The set A can consist some vertices of type u i or v i .Therefore, the following division is considered.Note that, r vertices can be belong to the inner cycle of generalized peterson graph Assume the vertices u 1 , • • • , u i−1 ∈ A, then: 1) If the vertex u i−1 wants to force the vertex u i , then it is necessary that v i−1 ∈ A.
2) If the vertex u i+1 wants to force the vertex u i , then it is necessary that v i+1 , u i+1 , u i+2 ∈ A.
3) If the vertex v i wants to force the vertex u i , then it is necessary that v i , v i−k , v i+k ∈ A. Therefor the best case for the color-change processing in the vertices of the outer cycle is that the vertex u i−1 forces the vertex u i .So, suppose {u 1 , u 2 , u 3 , u 4 } ⊆ A. This set can not change the color of all vertices.By a simple argument, we conclude that the set A be {u 1 , u 2 , u 3 • • • , u 8 }.

A comparison between zero forcing sets and dynamic monopolies
In the last section, we compare the zero forcing sets with another propagation concept of graph theory.This concept is dynamic monopoly.It is studied by Zaker in [17].It is obvious that each ZF S is a 1-dynamo.For τ = 1, there does not exist any resistant subgraph.So, each subgraph can be a candidate for a dynamo of graph.[17] A resistant subgraph of G means each subgraph K such that for each vertex Zaker proved that each dynamo of graph does not contain any resistant subgraph of it [17].So, it is satisfy for the ZFS.There exists one lower bound for Z(G) which is obtained from the following results about dynamos.We know that each ZF S is a 1-dynamo.So, we have the following corollary.Also, it is equality for path (Z(P t ) = 1).The characterization of all graphs that satisfy this bound will be interesting.

Definition 4 . 1 .
[17] By a threshold assignment for the vertices of G we mean any function τ : V (G) → N ∪ {0}.A subset of vertices D is said to be a τ -dynamic monopoly of G or simply τ -dynamo of G, if for some nonnegative integer k, the vertices of G can be partitioned into subsets D 0 , D 1 , ..., D k such that D 0 = D and for any i, 1 ≤ i ≤ k, the set D i consists of all vertices v which has at least τ (v) neighbors in D 0 ∪ . . .∪ D i−1 .Denote the smallest size of any τ -dynamo of G by dyn(G).

Example 4 . 2 .
We know Z(K n ) = n − 1 and Z(P n ) = 1.The ZFS of complete graph K n and path P n are 1-dynamo too.For the complete graph K 4 , dyn(K 4 ) = Z(K 4 ).It is an interesting question that for what graphs there exist this equality.In this example the subsets D 0 and D 1 are ZF S.

Theorem 4 . 4 .
[17] Let D be a dynamic monopoly of size k in G. Set H = G \ D and let t max be the maximum threshold among the vertices of H. Then:1) v∈H t(v) ≤ |E(G)| − |E(G[D])| − δ(G) + t max .2) v∈H t(v) ≤ |E(G)| provided that t(v) ≤ d G(v)for any vertex v ∈ H.
the vertex u ik is the only white neighbor of the black vertex u ik+1 and is forced by it.The vertex v n is the only white neighbor of the black vertex v (r−1)k+s and is forced by it.The vertex v k is the only white neighbor of the black vertex v n and is forced by it.For i = 2, • • • , r − 1, the vertex v ik is the only white neighbor of the black vertex v (i−1)k and is forced by it.Now, for i = 1, . .., r:The vertex u ik−1 is the only white neighbor of the black vertex u ik and is forced by it.Then the vertex v ik−1 is the only white neighbor of the black vertex u ik−1 and is forced by it.Also for each t < k, it is one counter, we have: The vertex u ik−t is the only white neighbor of the black vertex u ik−t+1 and is forced by it.the vertex v ik−t is the only white neighbor of the black vertex u ik−t and is forced by it.