In recent years, researchers have become increasingly interested in dynamic domination processes on graphs and eternal domination problems on graphs have been particularly well-studied (see the survey , for example). In the all-guards move model for eternal domination, a set of vertices are occupied by “guards” and the vertices occupied by guards form a dominating set on a graph. At each step, an unoccupied vertex is attacked and then each guard may remain at their current vertex or move along an edge to a neighbouring vertex. The guards aim to occupy a dominating set that contains the attacked vertex and such a movement of guards is said to “defend against an attack”. The eternal domination number of a graph, denoted is the minimum number of guards required to defend against any sequence of attacks, where the subscript and superscript indicate that all guards can move in response to an attack and the sequence of attacks is infinite. Given the complexity of determining the eternal domination number of a graph for the all-guards move model, recent work such as [1, 3, 4, 5, 10, 12, 2], has focused primarily on bounding or determining the parameter for particular classes of graphs.
In this paper, we extend the notion of eternal domination to that of eternal -domination in the most natural way: suppose at time , the guards occupy a set of vertices that form a -dominating set. At each time step an unoccupied vertex is attacked and every guard moves distance at most so that the guards occupy the vertices of a -dominating set that contains the attacked vertex. We note that for , the process is equivalent to the all-guards move model for eternal domination as described above. Hence, we focus on results for .
We present preliminary results in Section 2 that provide general bounds, the complexity of the associated decision problem, and determine the eternal -domination number exactly for some small classes of graphs. In Section 2.2, we show that the eternal -domination number of a graph is bounded above by the eternal -domination number of a spanning tree of the graph, which motivates us then to focus on trees in Section 3. The eternal domination number of a tree was characterized in  by using two reductions. We extend the concepts of these reductions to the eternal -domination model, providing reductions that, informally speaking, “trim branches” of trees in such a way that the change in the eternal -domination number is controlled. However, for the eternal -domination model such reductions are insufficient to characterize the eternal -domination number of all trees and we state some resulting open problems. In Section 4, we provide an upper bound for the eternal -domination number of a tree, which can be extended to an upper bound for the eternal -domination number of a tree. We also determine exactly, the eternal -domination number for perfect -ary trees. Since this paper introduces the concept of eternal -domination, we conclude with a series of open questions in Section 5.
We conclude this section with some formal definitions. Recall, in a graph , the open distance- neighbourhood of is , and the closed distance- neighbourhood of is .
Let be a graph and an integer. A set is a -dominating set if every vertex of is at most distance from a vertex in . The minimum cardinality of a -dominating set in graph is the -domination number, denoted .
Let be a graph. Let be the set of all -dominating sets of which have cardinality . Let . We will say transforms to if , , and for all , the closed neighbourhood. If we consider as the placement of all the guards in a -dominating set, then the set is a permissible movement of all the guards after some attack.
An eternal -dominating family of is a subset for some so that for every and each possible attack , there is a -dominating set so that and transforms to .
A set is an eternal -dominating set if it is a member of some eternal -dominating family. Eternal domination is a discrete time-process, so at each iteration of an “attack” the -dominating set transforms into some other set within the family.
The eternal -domination number of a graph , denoted , is the minimum for which an eternal -dominating family of exists. We use this notation to indicate that all guards are allowed to move a distance of at most .
2 Eternal -domination on general graphs
2.1 Preliminary Results
In this section, we relate the eternal -domination number to known graph parameters in order to obtain bounds, as well as determine the eternal -domination number for well-known families of graphs. We then determine the complexity of computing this number.
By definition, any eternal -dominating set is also a -dominating set, giving us the lower bound in Observation 4. However, we can also bound the eternal -dominating number of a graph by its -domination number:
For any graph and integer ,
To demonstrate the upper bound, consider a -dominating set where and on graph . For each , the maximum distance between any two vertices in is . Specifically, a guard is placed on an arbitrary vertex of for each . The guard will only ever occupy vertices of . Then, given an attack at a vertex in , guard can move to the attacked vertex and no other guard moves. We note that it is possible that the attacked vertex is within distance from multiple vertices in .
The bounds in Observation 4 are tight. It is easy to see that if graph has a universal vertex, then . However, it is worth noting that the difference between and can be arbitrarily large. Consider where each edge is subdivided times and call this graph . Then clearly , but .
Though the bounds of Observation 4 are tight, it is important to note that for some graphs, such as cycles, can be much smaller than .
For and ,
Observe can be decomposed into vertex-disjoint paths of length at most . Since a center vertex of a path of length at most will -dominate the path, .
Next we show that a minimum -dominating set is indeed a minimum eternal -dominating set. Place guards on the vertices of the minimum -dominating set described above. Suppose vertex is attacked and let be a vertex within distance of that contains a guard. The guard at moves distance to occupy and all other guards move exactly distance in the same direction.∎
As a consequence of Theorem 5, whenever a graph is Hamiltonian, we obtain the following upper bound, by simply considering the guards moving strictly along the Hamilton cycle.
Let be a Hamiltonian graph on vertices. Then for , .
Although it is easy to see that , we next show the eternal -domination number of a path is larger.
For and , .
Let where is adjacent to for . By partitioning the path into vertex-disjoint sub-paths, each of length at most and assigning one guard to each such sub-path, it is easy to see that guards will suffice to form an eternal -dominating family on , by using the reasoning following Observation 4.
Next we prove the lower bound. There must always be a guard within distance of . Thus a guard is always required to be located on the sub-path . We partition the graph into sub-paths of length at most . Let be the sub-path of length induced by vertices , for , and be the sub-path , should it exist.
For some fixed , assume the placement of the guards on are such that for each , each sub-path always contains at least 1 guard; i.e. every eternal -dominating set contains exactly one vertex from for each . Thus is the lowest-indexed sub-path that does not always contain a guard. Let time be the first time there is no guard in and assume an attack happens on this path. Furthermore, no guard from can move into since each lower-indexed path must have a guard on it at all times. If there are no guards in that can move to the attacked vertex in , then the guards do not form a -dominating set.
Thus, suppose a guard in moves to the attacked vertex.
If the guard that moves from the sub-path to leaves the sub-path without a guard, a guard from most move onto , and so on. If there is some sub-path such that the guard on that path cannot be replaced by a guard in then the guards do not form an eternal -dominating set, as the subsequent attack can occur within this sub-path and not be guarded. Otherwise, each guard in sub-path moves to sub-path , eventually leaving unable to be defended should an attack occur there at time . Thus at the end of each time step, there must be a guard in each : so at least guards are required and the result follows.∎
The previous results show that for some families of graphs, the eternal -domination number grows linearly with the order of the graph. On the other end of the spectrum, we can easily characterize graphs with eternal -domination number : if and only if the diameter of graph is at most .
An important question to ask when investigating a graph parameter is, how difficult is it to compute? To answer this, we will look at the relationship between the eternal -domination number of a graph and it’s graph power. For a graph , the power of the graph, , is formed by adding an edge whenever . Thus, if there exists a path in from to of length at most , then we will witness an edge .
If is a graph and , then
Let be an eternal -dominating set of for which . Suppose a sequence of attacks, occur. For each guard , let be the set of corresponding defending moves the guard makes, that is guard moves from to after attack . We now consider the eternal 1-domination of by using corresponding moves of the eternal -domination of . Place the guards in on the vertices of , since . Suppose in the same sequence of attacks occur on vertices .
For any guard and their sequence , moving from to in is permissible as these vertices have distance at most in , and thus will be adjacent in . Each guard , can use the same sequence of moves and still guard , thus .
Similarly, let be an eternal -dominating set of for which . Suppose a sequence of attacks, occur. For each guard , we define analogously as above. We then consider the eternal -domination of , using moves from the eternal -domination in . Place the guards in on the vertices of and consider the eternal -domination process, and again suppose in the vertices are attacked in that order. Any guard that moves from to after an attack in can also move from to in since these two vertices are adjacent in and thus are at most distance in . Thus, , giving the desired result. ∎
In  and the subsequent errata , it was shown that deciding if a set of vertices of a graph is an eternal domination set is in EXP. Thus, taken with Theorem 8 we have the following complexity result.
Let be a graph of order , a positive integer with and . Deciding if is an eternal -dominating set for is in EXP.
2.2 Using trees to bound
In this section, we provide insight to the vertices “covered” or “guarded” by a single guard or a pair of guards, and present bounds on the eternal -domination number for arbitrary graphs based on a partitioning into vertex-disjoint trees.
A subgraph of a graph is a retract of if there is a homomorphism from to so that for . The map is called a retraction and we note that since this is an edge-preserving map to an induced subgraph, the distance between any two vertices does not increase in the image.
Let be a retract of graph . Then .
Let be a retract of graph and be a retraction. We consider two parallel incidences of eternal -domination: one on and one on . The process on can be thought of as taking place on , as is an induced subgraph of . We will restrict the attacks in to vertices that are also in . Initially, if there is a guard at vertex , then we place a guard at vertex . If a guard in moves from to in response to an attack at vertex in , we observe that guard may move from to in in response to the attack. Thus, .∎
Below we consider another subgraph that will also prove useful in obtaining an upper bound for for an arbitrary graph .
Let be a graph, a positive integer and , then
where is the subgraph of obtained by deleting edge .
Let be a graph and . Consider, the graph with with edge removed. Let be an eternal -dominating set of of minimum cardinality and suppose a sequence of attacks, occur. For each guard , let be the set of corresponding defending moves the guard makes, that is after the attack , the guard moves from to .
Now place guards on the vertices of in and consider the same sequence of attacks . Each guard can respond appropriately, moving to after attack . Since every edge of is an edge of , the result follows.∎
From repeated applications of Lemma 11 we obtain the following.
Let be a graph and a spanning tree of , then
Theorem 12 suggests that understanding the eternal -domination process on trees is important, as it provides an upper bound on the eternal -domination number of a general graph. We next consider covering a graph with sub-trees with a particular structure, each of which can be guarded by a single guard. By cover, we mean that a guard is assigned to a particular sub-tree and they can respond to any sequence of attacks that occur within that particular sub-tree.
A -rooted tree with , is a rooted tree where the eccentricity of the root is at most .
Given a graph , we define a -rooted tree decomposition to be a partition of the vertices into sets for such that contains a spanning subgraph that is -rooted tree.
The -rooted tree decomposition number of a graph , denoted , is the minimum number of sets over all possible decompositions.
An easy bound comes from partitioning graph into -rooted trees as one guard can cover the vertices of a -rooted tree.
For any graph and
For some graphs, a better bound can be achieved by partitioning graph into -rooted trees and recognizing that guards can protect the vertices of each -rooted tree.
If is a -rooted tree for some , then .
Let be a -rooted tree with root . Initially place one guard at and the second guard at an arbitrary vertex, . After a vertex, is attacked, the guard at moves to and the other guard at moves to . The resulting -dominating set is equivalent to the original, and the guards can respond to attacks in this manner indefinitely.∎
For any graph and ,
In light of Theorem 12 and the fact that we can partition a graph into -rooted trees in order to find an upper bound for , the next two sections will focus on trees.
3 Eternal -domination on trees
In this section, we consider the conditions for which the eternal -domination number of a tree will equal or be one greater than that of a sub-tree in the aims of working towards determining for any tree .
In , the authors provide a linear-time algorithm for determining the eternal -domination number of a tree. Their algorithm consists of repeatedly applying two reductions, R1 and R2, which we restate here.
R1: Let be a vertex of incident to at least two leaves and to exactly one vertex of degree at least two. Delete all leaves incident to .
R2: Let be a vertex of degree two in such that is adjacent to exactly one leaf, . Delete both and .
If is the result of applying either R1 or R2 to tree , then . With an aim to characterize the eternal -domination number of trees for , Propositions 18 and 19 generalize the reductions of  to arbitrary . Figure 1 (a) and (b) provide a visualization of the sub-trees described in Propositions 18 and 19, respectively.
Let and be neighbours in and let be the component of induced by the deletion of edge , that contains . If every leaf in is within distance of and , then
It is easy to see :
guards on an eternal -dominating set of sub-tree and place an additional guard at . Whenever a vertex of is attacked, a guard moves from to the attacked vertex and the guards in move as they would if was attacked in . Whenever a vertex of is attacked, the guards currently occupying vertices of move to form an eternal -dominating set on that contains the attacked vertex and the guard in moves to (it is possible that there is no guard in , in which case there are two guards on and when the guards move in response to the attack, one of the two guards remains on ).
We next prove that . Assume and we will show by way of contradiction that this is not the case. Place the guards on . Let and be leaves of at distance from and distance from each other. Note that a guard must be on to ensure that and are -dominated. We will consider what the eternal -domination process looks like on and on a copy of .
First, any attack in corresponds to the guards moving as required, ensuring a guard is on after the attack is defended. In this placement of guards is the same, as vertices in do not require guards, as has a guard.
Assume a vertex on the path is attacked. In a guard on must move to the attacked vertex (note: if another guard moves to the attack, it must go through , so we can assume this is the guard that moves). We then need to replace that guard otherwise is not -dominated.
On the tree this is equivalent to two guards moving onto , since attacks in and guards on vertices of correspond attacks and guards on on .
So we have two situations to consider. The first is if in an attack occurs at a vertex of . In it corresponds to having two guards, but one guard is superfluous (as it is not really in this sub-tree).
The second situation is if in an attack occurs at a vertex of . Then a guard on moves within as required and either a guard moves from to , or if had more than one guard, at least one of these guards do not move.
In both situations, the attacks lead to a response in that required one less guard to defend that sub-tree. This means that any sequence of attacks in result in requiring less than guards to eternally -dominate, a contradiction, so .∎
A suspended -end-path in a graph is a path of length at least such that at least one endpoint of the path is a leaf and all internal vertices of the path have degree exactly . The next result generalizes the R2 reduction in  for eternal -domination. See Figure 1 (b).
Suppose contains a suspended -end-path and label the vertices of as where is a leaf and is adjacent to for . Then
Clearly and since it is a path of length . We next show that guards do not suffice to eternally -dominate .
First, we consider eternal -domination on the graph . Place the guards and consider a minimal finite sequence of attacks that requires all guards to defend this sequence of attacks. That is, if there is at least one less guard, this sequence of attacks is not able to be defended.
Now consider the graph . Suppose and place the guards on the vertices of an eternal -dominating set of , note that there is at least one guard on . Consider the sequence of attacks , on tree . Thus there is always a guard in . This means that guard is never able to defend a vertex in , hence, there are not enough guards to eternally -dominate , a contradiction.∎
The two previous results provided a means to “trim branches” off a tree to reduce the eternal -domination number by . In each of these propositions, the diameter of is either exactly or . When , there are more interesting interactions between the guards in and in . It may be possible for guards to move in and out of while guarding , so may or may not decrease by one. In fact, for every diameter in , we next provide a construction where there exists a tree with and a tree with .
We first consider a tree , rooted at a vertex , with one leaf at distance from , with for some . In our first example, to define we consider and add a star centered at a vertex , with each leaf at distance from , and add the edge . In this case, , since a guard on a leaf in the star can move to cover , and we can ensure a guard occupies at all times. Then, whenever there is an attack on the leaf at distance from , and then a subsequent attack on the vertex in at distance from this leaf, we require the guard on to move into t the new attack, and the previous guard can move back up to .
For the second example, we define a tree by considering and adding star centered at a vertex , with each leaf at distance from , and add the edge . In this case, . A guard on the leaf of the star will not be able to guard , and we have one guard always on in , however, in order to guard the leaf at distance from in , we require a third guard in in order to ensure there is always a guard on available to guard the leaves as above.
Although the previous results provide insight into a reduction on trees, the two examples above show that additional conditions are needed to characterize sub-trees with diameters between and .
Our next result provides a way to “trim branches” without changing the eternal -domination number. For , for example, if a vertex is adjacent to at least two leaves, then a leaf can be deleted without changing the eternal -domination number. An example of Theorem 22 having been applied seven times to a tree is shown in Figure 4.
Let be an induced subgraph of a tree where is a -rooted tree with root , some leaf distance from in , and is connected. Then where is the set of vertices on the -path.
By Lemma 10, . It is easy to see that guards suffice to eternally -dominate . The guards on move as they would on graph , with a few exceptions: when a vertex of is attacked in , the guards of move to the same vertices guards of would move to in response to an attack at a vertex on , with the exception of one guard who moves to the attacked vertex on , rather than a vertex on . ∎
Lemma 23 describes another set of vertices that can be deleted from a tree without changing the eternal -domination number. We first identify two leaves that are distance apart and let be the centre of the -path. Informally, we identify all “branches” from whose leaves are all within distance of and remove all vertices on these branches, apart from those on the -path. Lemma 23 proves the sub-tree has the same eternal -domination number as the original tree. An example is shown in Figure 2, where the vertices removed (defined as set in the theorem) are striped.
The eccentricity of vertex in graph , denoted , is the maximum distance between and any other vertex in . More formally, .
Suppose are two leaves in tree such that . Let denote the -path and let be the center of .
Let for some integer . Let be the components of induced by the deletion of where . Suppose for and for . If
By Theorem 10, . Suppose . We simply modify the movements of the guards on to defend against attacks on vertices of in .
We first point out the existence of a particular eternal -dominating family on . Each eternal -dominating set on must contain at least one vertex on the -path (otherwise is not -dominated). Suppose that in response to an attack at a vertex on the -path, the guards move to form a -dominating set that does not contain . Clearly, the guards could alternately have moved to form a -dominating set that does contain : after the guards move in response to the attack, there must be a guard on the -path (otherwise is not -dominated) this guard could have moved to . Thus, there exists an eternal -dominating family on where each eternal -dominating set is of cardinality and contains . We now exploit this eternal -dominating family in order to create an eternal -dominating family for of cardinality .
Initially, place guards on the vertices of that correspond to the vertices of some eternal -dominating set on . Since this results in a guard on vertex on , we know is initially -dominated. If a vertex is attacked in , the guards of mirror the movements of guards in graph in response to an attack at . If a guard in graph moves from vertex to , then a guard in graph will move from vertex to , with one exception: if there is a guard in , the guard will move to . Note that in , this results in the vertices of remaining -dominated.
Now suppose that on , vertex is attacked. If the previous vertex attacked was also in , then a guard moves from to and a guard moves from to . Otherwise, we consider an attack at in graph . On graph , a guard can move from to and the remaining guards move accordingly, to form an eternal -dominating set containing both the attacked vertex and . In , a guard moves from to instead of and the remaining guards move the same as their counterparts in . Thus, the guards on form a -dominating set.∎
For , we next consider two examples which illustrate that for some trees, removing will not change the eternal -domination number, but for other trees, it will. Thus, Examples 24 and 25 illustrate that Lemma 23 cannot be improved by removing vertices of .
For , consider the tree given in Figure 3 (a), where vertex has been identified. Using Lemma 23, we can “trim branches” of the tree without changing the eternal -domination number. The tree shown in Figure 3 (b) shows the result of applying Lemma 23 to tree using the vertex identified as in the figure. We note that though , it is also the case that . (In Figure 3 (a) and (b), this is the part of the graph outside of the bubble). Thus, for this example, “trimming off” does not change the eternal -domination number.
In Section 4.2, we present further tree reductions for the case where .
4 Eternal -domination on trees
4.1 General Results for Trees and -ary Trees
In this section, we first describe an eternal -dominating set for any tree , which yields an upper bound for , and second, determine exactly when is a perfect -ary tree.
Let be a rooted tree and place guards according to 1.-3. below. The vertices occupied by guards form an eternal -dominating set on .
Initially place a guard on each vertex for which the distance to the nearest leaf is even and positive.
If 1. results in no guards being placed on the root, then place one guard on the root.
Place one guard on an arbitrary leaf.
It is easy to see that the result holds for any rooted tree of depth or . We prove the result by inducting on the depth of the tree. Let be a tree of depth . Let be the sub-tree of depth induced by the deletion all leaves of to get ; then delete all leaves of to get (note: leaves of are the stems in with at most one non-leaf neighbour).
The map above preserves distances in the sub-tree and maps leaves in and to the nearest vertex that is distance from a leaf, noting that such a vertex may also happen to be a leaf of .
Let be any eternal -dominating set on . We create a -dominating set on that “mirrors” on . Let be the set of vertices on where
(a) if then is in ; and
(b) for each vertex of that is a grandparent of a leaf, if then and if , then choose an arbitrary child of to be in .
From this construction, is a -dominating set on .
Suppose vertex is attacked. We consider two cases: (1) when is neither a leaf in nor a leaf in and (2) when is a leaf in or .
(1) Suppose is neither a leaf in nor leaf in . Then . In this situation, we consider an attack at vertex in . The guards of move from an eternal -dominating set to an eternal -dominating set that contains . We move the guards in according to how the guards moves in . That is, if a guard in moves from to , then in the guard at moves . Additionally, any guard in or that is located at a leaf (that is not already a grandparent of a leaf) moves to the nearest vertex that is a grandparent of a leaf. Observe that . Let be the set of vertices now occupied by guards in . Observe that is a -dominating set on that contains . Further observe that mirrors , just as mirrored .
(2) Suppose is a leaf in or . Then for where and is the nearest (to ) grandparent of a leaf in .
If there is a guard on a vertex that is a child or grandchild of , then this guard moves to while the guard at moves to the attacked vertex. Call the set of vertices now occupied by the guards and note that it is a -dominating set containing the attacked vertex. Observe that mirrors just as mirrored .
Next, suppose there is no guard on a vertex that is child or grandchild of . Then we consider an attack at and the resulting movements of guards in . In , suppose the guards move from to the the eternal -dominating set that contains .
If a guard in moves from to , then the guard at moves . The guard at moves to the attacked vertex. Finally, any there is a guard on a vertex that is a child or grandchild of where and is the grandparent of a leaf, then that guard moves to . Call the set of vertices now occupied by the guards and note that it is a -dominating set containing the attacked vertex. Observe that mirrors just as mirrored .
We have seen that for any attack, the movements of guard on can be guided by the movements of guards on sub-tree . After each attack, if the guards on can move to an eternal -dominating set , then the guards on can move to a -dominating set . ∎
We note that although the result of Lemma 26 is expressed for , the result and proof can easily be extended to arbitrary :
Initially place a guard on each vertex for which the distance to the nearest leaf is a positive multiple of .
If 1. results in no guards being placed on the root, then place one guard on the root.
Place one guard on an arbitrary leaf.
In this paper, we only use Lemma 26 for the case, so we do not prove the result for arbitrary . However, combining Lemma 26 or the above extension with Theorem 12, will yield an upper bound for the eternal -domination number (or eternal -domination number) of any graph.
An -ary tree is a rooted tree where every vertex has at most children. The depth of an -ary tree is the eccentricity of the root. A perfect -ary tree is an -ary tree in which every non-leaf vertex has exactly children and every leaf is distance from the root where denotes the depth of the tree.
We next present a lemma that will be helpful in determining the eternal -domination number for perfect -ary trees. Recall that an eternal -dominating family is a set in which the elements are eternal -dominating sets, all of the same cardinality, so that: if the guards occupy eternal -dominating set and there is an attack at vertex , the guards can move from set to a set that contains (i.e. and transforms to ). A minimal eternal -dominating family is minimal in terms of the number of eternal -dominating sets in the family.
Let be a perfect -ary tree of depth for . There exists a minimal eternal -dominating family in which each eternal -dominating set contains the grandparent of every leaf and has cardinality