Abstract
An eternal dominating set of a graph 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 -eternal domination model. The size of the smallest -eternal dominating set is called the -eternal domination number and is denoted by . In this paper, we find and for . We also find upper bounds for and when is arbitrary.
1. 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 [1–5] 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 were determined in [3], where denotes the independence number of and denotes the domination number of . The -eternal domination numbers for cycles and paths were found by Goddard et al. [3] as follows: and . For further information on eternal domination, see survey [1]. A generalized Petersen graph is a graph with vertex set and edge set with , , and and ; see [6] for more information on generalized Petersen graph. The -dominating graph was defined by Goldwasser et al. in [7] as follows. Let be a graph with a dominating set of cardinality . The vertex set of the -dominating graph , denoted , is the set of all subsets of of size which are dominating sets and two vertices of are adjacent if and only if the guards occupying the vertices of of one can move (at most distance one each) to the vertices of the other, if and only if has an induced subgraph such that for each vertex of the union of the vertices in the closed neighborhood of in is equal to . A generalized Jahangir graph for is a graph on vertices, i.e., a graph consisting of a cycle with one additional vertex which is adjacent to vertices of at distance from each other on ; see [8] for more information on the Jahangir graph. In [9], we found for . In [10], we found for are arbitraries.
Proposition 1 [11]. For , we have
Proposition 2 [11]. For , we have
Proposition 3 [11]. For , we have
Proposition 4 [12]. For , we have
Proposition 5 [12]. For , we have
Proposition 6 [3]. For any graph .
2. Main Results
2.1. Eternal Domination Number of
In this section, we find the -eternal domination number of for is arbitrary.
Theorem 1.
Proof. We consider all four cases of .
Case 1 (). Let . As it was found in [11] that for this specific case and the -dominating set is the union of two sets and which are ,
We form the -dominating graph where , , and with sets
Each of these sets has a cardinality of and they are adjacent in for the following reasons.
, , , by rotating the guards one step clockwise.
, , , by rotating the guards one step counterclockwise.
, , , by reflection which means moving the guards one step to occupy the corresponding unoccupied vertex on the other cycle.
Therefore, are adjacent in (the induced subgraph of ) on and ; therefore, we conclude that but from the definition of eternal domination; therefore, for ; see Figure 1.
Case 2 (). Let . It was found in [11] that for this case and it was found that the -dominating set is .
We form the -dominating graph with the following sets of cardinality:
It is easy to notice that are all adjacent in and the union ; therefore, for but we have from the definition of eternal domination; therefore, for ; see Figure 2.
Case 3 (). Let . As it was found in [11], for this specific case and the -dominating set is .
We form the -dominating graph with the following sets of the cardinality :
It is easy to notice that are all adjacent in and the union of sets ; therefore, for , ; therefore, ; see Figure 3.
Case 4 (). Let . Ebrahimi et al. found in [11] that for this case and the -dominating set is .
We form the -dominating graph with the following sets of cardinality :
It is easy to notice that are all adjacent in and the union ; therefore, for but ; therefore, for ; see Figure 4. From Cases 1–4, we conclude that for .
2.2. Eternal Domination Number of
In this section, we give an upper bound for the eternal domination number of generalized Petersen graph .
Lemma 1. If we divide into blocks where each block consists of 10 vertices (5 vertices from the outer cycle and their 5 adjacent vertices from the inner cycle), then 4 guards are enough to eternally dominate each block.
Proof. We denote these blocks by ; we also denote the subblock that remains in case by keeping in mind that consists of 2, 4, 6, and 8 vertices when we have , respectively. It was found in [11] that three vertices are enough to dominate each block if the dominating vertices were distributed as shown in Figure 5.
To prove that four guards can eternally protect any block it is enough to find a set of configurations for the guards so that these configurations cover all the vertices of (every vertex of 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 -dominating graph, we conclude that .
Theorem 2. For ,
Proof. We discuss all the cases as follows.
Case i.1 ().
From Propositions 2, 4, and 6, we conclude that
By forming (the 3-dominating graph for ), we notice that consists of the following sets:
Figure 7 illustrates .
We denote the closed neighborhood of vertex in by , and we notice the following:
Therefore, by the definition of -dominating graph, we conclude that which means from (6).
We can easily prove that the following family is a 4-vertex eternal dominating family of Case i.2 ().
From Propositions 2, 4, and 6, we conclude that which means and the following family is a 4-vertex eternal dominating family of Case ii ().
From Propositions 2, 4, and 6, we conclude that . The following family is a 5-vertex eternal dominating family of :
Case iii ().
From Propositions 2, 4, and 6, it can be implied that
Case iv ().
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 ; 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 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 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 which is the middle “outer” vertex of and then the guards of return to configuration 1 of Figure 7 and at the same time the guards of , move to configuration 3 of Figure 7. It is obvious that these configurations can be applied to all sets of blocks in which means without loss of generality the theory holds and we conclude that in this case.
Case v ().
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 against an endless series of attacks taking into consideration the need to assign 4 guards to in order to eternally protect it without the need to summon a guard from another block and that is because 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.
Case vi ().
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 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 , 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 and the additional guards on the subblock SB so that these configurations can cover all the vertices of 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 guards when .
Case vii ().
This case is similar to Case vi with the exception that we need one additional guard on because we have an odd number of blocks (similar to Case v).
Case viii ().
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 ().
This case is similar to Case viii with the exception that we need one additional guard on because we have an odd number of blocks (similar to Cases v and vii).
We conclude that the theorem holds for all cases of .
2.3. Eternal Domination Number of
In this section, we find the eternal domination number of generalized Petersen graph when and we give an upper bound for in the remaining cases: .
Theorem 3. for .
Proof. Let . From Proposition 3, we know that for this case and the -dominating set is the union of two sets and which are , . Therefore, according to Propositions 3 and 5, the trivial bounds for are .
We form the -dominating graph where ,, and with sets
Each of these sets has a cardinality of and they are all adjacent in for the following reasons.
, , , by rotating the guards of the outer cycle one step clockwise and rotating the guards of the inner cycle one step counterclockwise.
, , , by rotating the guards on the outer cycle one step counterclockwise and rotating the guards of the inner cycle one step clockwise.
, , , by reflection which means moving the guards from each cycle one step to occupy the corresponding unoccupied vertex on the other cycle.
Therefore, are adjacent in (the induced subgraph of ) on and ; therefore, we conclude that for but from the definition of eternal domination, so for ; see Figure 13.
This proof proposes the following idea.
For , we can divide into two sets and defined as
We notice that any change of equals a change in the number of the vertices of ; meanwhile, the number of the vertices of does not change. We can use this observation to study the eternal domination problem on when , because this observation allows us to take the relationships between the guards situated on the vertices of and the end vertices of of one example “one value of ” then generalize these relationships for any value of , taking into consideration that the relationships between the guards situated on the inner vertices of will always be similar to the ones in Theorem 3.
Theorem 4. For , we have
Case 1 ( and ). Let . From Proposition 3, we know that for this case and the -dominating set is . Therefore, according to Propositions 3 and 5, the trivial bounds for are .
We form the -dominating graph with these sets of cardinality:
It is easy to notice that are all adjacent in and the union ; therefore, for . See Figure 14.
Case 2 ( and ). Let . From Proposition 3, we know that for this case and the -dominating set is . Then, according to Propositions 3 and 5, the trivial bounds for in this case are .
We form the -dominating graph with these sets of cardinality:
See Figure 15.
Case 3 ( and ). Let . From Proposition 3, we know that for this case and the -dominating set is . Therefore, according to Propositions 3 and 5, the trivial bounds for in this specific case are .
We form the -dominating graph with these sets of cardinality:
It is easy to notice that are all adjacent in and the union ; therefore, for . See Figure 16.
From Cases 1–3, we conclude the requested.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Faculty of Science, Tishreen University, Syria.