Eternal Domination of Generalized Petersen Graph

An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks; an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the m-eternal domination model. The size of the smallest m-eternal dominating set is called the m-eternal domination number and is denoted by γm ðGÞ. In this paper, we find γm ðPðn, 1ÞÞ and γm ðPðn, 3ÞÞ for n ≡ 0 ðmod 4Þ. We also find upper bounds for γm ðPðn, 2ÞÞ and γm ðPðn, 3ÞÞ when n is arbitrary.


Introduction
The term graph protection refers to the process of placing guards or mobile agents in order to defend against a sequence of attacks on a network. Go to [1][2][3][4][5] for more background of the graph protection problem. Burger et al. introduced the concept of eternal domination in 2004 [2]. Goddard et al. introduced the "all guards move model" in [3]. General bounds of γðGÞ ≤ γ ∞ m ðGÞ ≤ αðGÞ were determined in [3], where αðGÞ denotes the independence number of G and γðGÞ denotes the domination number of G. The m-eternal domination numbers for cycles C n and paths P n were found by Goddard et al. [3] as follows: γ ∞ m ðC n Þ = dn/3e and γ ∞ m ðP n Þ = dn/2e. For further information on eternal domination, see survey [1]. A generalized Petersen graph Pðn, kÞ is a graph with vertex set V ∪ U = fv 1 , v 2 , ⋯, v n g ∪ fu 1 , u 2 , ⋯, u n g and edge set E = ∪fv i v i+1 , v i u 1 , u i u i+k g with v n+1 = v 1 , u n+1 = u 1 , and 1 ≤ i ≤ n and 1 ≤ k ≤ bn/2c; see [6] for more information on generalized Petersen graph. The k-dominating graph HðG, kÞ was defined by Goldwasser et al. in [7] as follows. Let G be a graph with a dominating set of cardinality k.
The vertex set of the k-dominating graph HðG, kÞ, denoted VðHÞ, is the set of all subsets of VðGÞ of size k which are dominating sets and two vertices of H are adjacent if and only if the k guards occupying the vertices of G of one can move (at most distance one each) to the vertices of the other, γ ∞ m ðGÞ ≤ k if and only if HðG, kÞ has an induced subgraph SðG, kÞ such that for each vertex x of SðG, kÞ the union of the vertices in the closed neighborhood of x in S ðG, kÞ is equal to VðGÞ. A generalized Jahangir graph J s,m for m ≥ 2 is a graph on sm + 1 vertices, i.e., a graph consisting of a cycle C sm with one additional vertex which is adjacent to m vertices of C sm at distance s from each other on C sm ; see [8] for more information on the Jahangir graph. In [9], we found γ ∞ m ðJ s,m Þ for s ≡ 2, 3. In [10], we found γ ∞ m ðJ s,m Þ for s, m are arbitraries. Proposition 2 [11]. For n ≥ 5, we have γðPðn, 2ÞÞ = d3n/5e: Proposition 3 [11]. For n ≥ 7, we have γðPðn, 3ÞÞ = ðn/2Þ + 1 for n ≡ 2ðmod 4Þ, dn/2e for n ≡ 0, 1ðmod 4Þ or n = 11, dn/2e + 1 for n ≡ 3ðmod 4Þ, n ≠ 11: Proposition 4 [12]. For n ≥ 5, we have αðPðn, 2ÞÞ = b4n/5c: Proposition 5 [12]. For n ≥ 7, we have αðPðn, 3ÞÞ = n if n is even, n − 2 if n is odd: ( Proposition 6 [3]. For any graph G :γðGÞ ≤ γ ∞ m ðGÞ ≤ αðGÞ. ( Proof. We consider all four cases of n.

Main Results
Case 1 (n ≡ 0ðmod 4Þ). Let n = 4l : l ∈ N. As it was found in [11] that γðPðn, 1ÞÞ = n/2 for this specific case and the γ-dominating set S is the union of two sets A and B which are We form the k-dominating graph HðG, kÞ where G = Pðn, 1Þ, k = n/2, and with sets Each of these sets D 1 , D 2 , D 3 , D 4 has a cardinality of n/2 and they are adjacent in HðPðn, 1Þ, n/2Þ for the following reasons.
by rotating the guards one step clockwise.
by rotating the guards one step counterclockwise.
Case 2 (n ≡ 1 ðmod 4Þ). Let n = 4l + 1 : l ∈ N. It was found in [11] that γðPðn, 1ÞÞ = dn/2e for this case and it was found that the γ-dominating set S is S = A ∪ B ∪ fx g: x ∈ fv n , u n g.

Eternal Domination
Proof. We denote these blocks by B j : 1 ≤ j ≤ bn/5c; we also denote the subblock that remains in case n ≢ 0 ðmod 5Þ by SB keeping in mind that SB consists of 2, 4, 6, and 8 vertices when we have n ≡ 1, 2, 3, 4 ðmod 5Þ, respectively. It was found in [11] that three vertices are enough to dominate each block B j if the dominating vertices were distributed as shown in Figure 5.
To prove that four guards can eternally protect any block B j it is enough to find a set of configurations for the guards so that these configurations cover all the vertices of B j (every vertex of B j belongs to at least one configuration) and every two configurations are reachable from each other in one step; we notice that the configurations presented in Figure 6 can achieve that; therefore, according to the definition of the k-dominating graph, we conclude that γ ∞ m ðB j Þ ≤ 4.
In this case, we have an even number of blocks that were defined in Lemma 1. For every pair of adjacent blocks, let us assign 3 guards to one of them and 4 guards to the other; we start by studying the first two blocks B 1 , B 2 ; to prove that the 7 guards assigned to these two blocks are enough to eternally protect them, it is enough to find a set of guard configurations for which every vertex of VðB 1 Þ ∪ VðB 2 Þ belongs to at least one of the configurations and every two configurations are reachable from one another in one step. We imply that the configurations shown in Figure 8 satisfy this purpose.
Therefore, the number of guards we assigned to B 1 , B 2 is enough to eternally protect these two blocks. Let us now consider the next attack to occur on a vertex that does not belong to any of these two blocks; for example, the vertex denoted by v 18 which is the middle "outer" vertex of B 4 and then the guards of B 1 , B 2 return to configuration 1 of Figure 7 and at the same time the guards of B 3 , B 4 , B 5 , B 6 move to configuration 3 of Figure 7. It is obvious that these configurations can be applied to all sets of blocks in Pðn, 2Þ which means without loss of generality the theory holds and we conclude that γ ∞ m ðPðn, 2ÞÞ ≤ ð3n/5Þ + dn/10e in this case.
In this case, we have an odd number of blocks that were defined in Lemma 1. This proof is equivalent to proving that the same distribution in Figure 7 of Case iv can eternally protect Pðn, 2Þ against an endless series of attacks taking into consideration the need to assign 4 guards to B bn/5c in order to eternally protect it without the need to summon a guard from another block and that is because B bn/5c does not belong to a pair (we have an odd number of blocks), which explains the need to have one additional guard in this case over Case iv.
In this case, we have an even number of blocks that were defined in Lemma 1. This proof is equivalent to proving that the same distribution in Figure 7 of Case iv can eternally protect Pðn, 2Þ against an endless series of attacks with one difference which is placing 1,2,3 additional guards on the additional 2,4,6-vertex subblock when n ≡ 1, 2, 3ðmod 5Þ, respectively. In each of Figures 9-11 and in a similar way to Case iv, we find a set of configurations of the 7 guards placed on B 1 , B 2 and the additional guards on the subblock SB so that these configurations can cover all the vertices of B 1 , B 2 , SB and they are all reachable from each other in one step.
As for the remaining pairs of blocks, we can simply apply the same configurations shown in Figure 8 of Case iv. Therefore, the theory holds for all the blocks and we prove the need for b3n/5c + dn/10e + l − 1 guards when l = 1, 2, 3.
This case is similar to Case vi with the exception that we need one additional guard on B bn/5c because we have an odd number of blocks (similar to Case v).
Case viii (n ≡ 4 ðmod 10Þ). This case is similar to Case vi, with the exception that we can eternally protect the 8-vertex subblock SB with l-2 = 2 guards instead of l-1 guards like in Case vi. Figure 12 shows the distributions of guards in this case.
Case ix (n ≡ 4 ðmod 5Þ and n ≢ 4ðmod 10Þ). This case is similar to Case viii with the exception that we need one additional guard on B bn/5c because we have an odd number of blocks (similar to Cases v and vii).
We conclude that the theorem holds for all cases of n.

Eternal Domination Number of Pðn, 3Þ.
In this section, we find the eternal domination number of generalized Petersen graph Pðn, 3Þ when n ≡ 0 ðmod 4Þ and we give an upper bound for γ ∞ m ðPðn, 3ÞÞ in the remaining cases: n ≡ 1, 2, 3 ðmod 4Þ. Proof. Let n = 4l : l ∈ N. From Proposition 3, we know that γðPðn, 3ÞÞ = n/2 for this case and the γ-dominating set S is the union of two sets A and B which are A = fv 4i+1 ; 0 ≤ i ≤ l − 1g, B = fu 4i+3 ; 0 ≤ i ≤ l − 1g. Therefore, according to Propositions 3 and 5, the trivial bounds for γ ∞ m ðPðn, 3ÞÞ are n/2 ≤ γ ∞ m ðPðn, 3ÞÞ ≤ n. We form the k-dominating graph HðG, kÞ where G = Pðn, 3Þ, k = n/2, and with sets Each of these sets D 1 , D 2 , D 3 , D 4 has a cardinality of n/2 and they are all adjacent in HðPðn, 3Þ, n/2Þ for the following reasons.