Eternal Independent Sets in Graphs

The use of mobile guards to protect a graph has received much attention in the literature of late in the form of eternal dominating sets, eternal vertex covers and other models of graph protection. In this paper, eternal independent sets are introduced. These are independent sets such that the following can be iterated forever: a vertex in the independent set can be replaced with a neighboring vertex and the resulting set is independent.


Graph Protection
Let G = (V, E) denote a finite, undirected graph with vertex set V and edge set E. The problem of protecting a graph with mobile guards has been studied in a number of recent papers. We shall begin with a review of some of these models before introducing the eternal independent set problem, which can be viewed in the same light.
A dominating set of a graph G = (V, E) is a set D ⊆ V such that each vertex in V − D is adjacent to a vertex in D. The minimum cardinality amongst all dominating sets of G is the domination number γ(G).
Let {D i }, D i ⊆ V , i ≥ 1, be a collection of sets of vertices of the same cardinality, with one guard located on each vertex of D i . Each protection problem can be modeled as a two-player game between a defender and an attacker : the defender chooses D 1 as well as each D i , i > 1, while the attacker chooses the locations of the attacks r 1 , r 2 , . . . (which are sometimes called requests). Each attack is dealt with by the defender by choosing the next D i in response to the attack r i , subject to some constraints that depend on the particular game. The defender wins the game if they can successfully defend any sequence of attacks, subject to the constraints of the game described below; the attacker wins otherwise. We note that the sequence of attacks may be infinite in length.
We say that a vertex (edge) is protected if there is a guard on the vertex or on an adjacent (incident) vertex. A vertex v is occupied if there is a guard on v, otherwise v is unoccupied. An attack is defended if a guard moves to the attacked vertex (across one edge, i.e., in one "step").

Eternal Protection Problems
For the eternal domination problem, each D i , i ≥ 1, is required to be a dominating set, r i ∈ V (assume without loss of generality r i / ∈ D i ), and D i+1 is obtained from D i by moving one guard to r i from an adjacent vertex v ∈ D i . If the defender can win the game with the sets {D i }, then each D i is an eternal dominating set. The size of a smallest eternal dominating set of G is the eternal domination number γ ∞ (G). This problem was first studied by Burger et al. in [1] and will be referred to as the one-guard moves model.
For the m-eternal dominating set problem, each D i , i ≥ 1, is required to be a dominating set, r i ∈ V (assume without loss of generality r i / ∈ D i ), and D i+1 is obtained from D i by allowing each guard to move to a neighboring vertex (if it so chooses). That is, each guard in D i may move to an adjacent vertex, as long as one guard moves to r i . Thus it is required that r i ∈ D i+1 . The size of a smallest m-eternal dominating set (defined similar to an eternal dominating set) of G is the m-eternal domination number γ ∞ m (G). This "all guards move" version of the problem was introduced by Goddard, Hedetniemi and Hedetniemi [3]. The m in m-eternal denotes that multiple guards may move in response to an attack.
In the eviction model, each configuration D i , i ≥ 1, of guards is required to be a dominating set. An attack occurs at a vertex r i ∈ D i such that there exists at least one v ∈ N (r i ) with v / ∈ D i . The next guard configuration D i+1 is obtained from D i by moving the guard from r i to a vertex v ∈ N (r i ), v / ∈ D i (i.e., this is the "one-guard moves" model). The size of a smallest eternal dominating set in the eviction model for G is denoted e ∞ (G). That is, attacks occur at vertices with guards and we must move that guard to an unoccupied neighboring vertex. This problem was introduced in [6].
A vertex cover of G is a set C ⊆ V such that each edge of G is incident with a vertex in C. The minimum cardinality of a vertex cover of G is the vertex cover number τ (G) of G. An independent set of G is a set I ⊆ V such that no two vertices in I are adjacent. The maximum cardinality amongst all independent sets is the independence number α(G). It is well known that α(G) + τ (G) = n for all graphs G of order n (see e.g. [2, p. 241]).
The clique covering number θ(G) is the minimum number k of sets in a partition V = V 1 ∪ · · · ∪ V k of V such that each G[V i ] is complete. Hence, as is well-known, θ(G) equals the chromatic number χ(G) of the complement G of G. Thus for every graph G, α(G) ≤ θ(G).
A matching in G is a set of edges, no two of which have a common endvertex. The matching number m(G) is the maximum cardinality of a matching of G. It is also well known that τ (G) ≥ m(G) for all graphs, and that equality holds for bipartite graphs (see e.g. For the m-eternal vertex covering problem, each D i , i ≥ 1, is required to be a vertex cover, r i ∈ E, and D i+1 is obtained from D i by moving one or more guards to neighboring vertices; i.e., each guard in D i may move to an adjacent vertex provided that one guard moves across edge r i (we assume without loss of generality that one end-vertex of r i is not in D i , otherwise the two guards on the endvertices of r i simply interchange positions). If the defender can win the game with the sets {D i }, then each D i is an eternal vertex cover. The size of a smallest eternal vertex cover of G is the eternal covering number τ ∞ m (G). This problem was introduced in [7].
A survey on eternal protection problems can be found in [8].

Eternal Independent Sets
For the eternal independent set problem each D i , i ≥ 1, is required to be an independent set, r i ∈ D i , and D i+1 is obtained from D i by moving the guard on r i to an adjacent vertex. (We say the vertex r i is attacked.) If the defender can win the game with the sets {D i }, then each D i is an eternal independent set. The size of a largest eternal independent set of G is the eternal independence number α ∞ (G). This will sometimes be referred to as the one-guard moves model.
For the m-eternal independent set problem each D i , i ≥ 1, is required to be an independent set, r i ∈ D i , and D i+1 is obtained from D i by moving the guard on r i to an adjacent vertex while the remaining guards in D i may also move to neighboring vertices (so long as r i / ∈ D i+1 ). If the defender can win the game with the sets {D i }, then each D i is an m-eternal independent set. The size of a largest m-eternal independent set of G is the m-eternal independence number α ∞ m (G). This will sometimes be referred to as the all-guards move model.
We shall sometimes say that if a guard at u moves to v, that vertices u and v are switched. A total-switch of an independent set D into an independent set Z is a simultaneous replacement of all vertices in D where each vertex v i ∈ D is replaced by a neighbor z i such that |D| = |Z|. Note that D ∩ Z = ∅, since D is an independent set. For the total-eternal independent set problem each D i , i ≥ 1, is required to be an independent set, r i ∈ D i , and D i+1 is obtained from D i by a total-switch. If the defender can win the game with the sets {D i }, then each D i is an total-eternal independent set. The largest cardinality of a total-eternal independent set is the total-eternal independence number of G denoted α ∞ t (G).
Observe that for the total-eternal independent set problem, the actual sequence of attacks does not matter, since all the guards must move upon each attack.
These eternal independent set problems are analogous to the eviction model of eternal domination. Related concepts for independent sets have been considered in [4,5,9], but the exact parameters defined here have not been studied prior to this, as far as we know.

Examples
We give a few small examples to illustrate the various definition. Observe that α ∞ ( We alert the reader to the fact that C 5 is an example that will be used several more times throughout the paper and illustrated in Figure  1. In Figure 1, a guard on a shaded vertex can move to an unshaded neighbor (the left guard must move clockwise, the right guard must move counterclockwise from this initial configuration) and the resulting guard configuration induces an independent set (and is isomorphic to the initial configuration). Also α ∞ (K n,n ) = 1 and α ∞ m (K n,n ) = n = α ∞ t (K n,n ). The corona of a graph G, denoted cor(G), is the graph obtained from G by adding a pendant vertex to every vertex of G. cor( More generally, it is easy to see that α ∞ (cor(G)) = α(G) and the simple proof is omitted.
follows from the second part of the proof of Theorem 4.2, below.

Clearly
is true follows since we can perform a total-switch along the edges of a free-matching. In a total-switch request for D, we replace D by Z where v i moves to u i . If another request is done (now on Z) we switch back to D. Hence D is total-eternal and α ∞ t (G) ≥ |D| = m f (G). Conversely, let D be a maximum total eternal independent set. A total-switch sends D to Z such that Z is independent, |D| = |Z| and every vertex v i in D moved to a neighbor z i in Z. Since |Z| = |D|, it follows that every v i moved to a distinct neighbor z i in Z hence Then v 1 moves to a neighbor z 1 , v 2 to a neighbor z 2 , and so on so Z = {z 1 , . . . , z k } is an independent set. Now a total-switch of D sends it to Z via the same edges e i = (v i , z i ) and a total switch on Z send it back to D, hence D is also total eternal independent set, and α ∞ t (G) ≥ |D| = α ∞ (G). In any infinite sequence of switchings imposed on D, we always keep moving v i to u i and u i to v i . So all these requests keep us with independent set T with none/some/all vertices in D and none/some/all vertices in Z such that |D| = |T | = |Z|; hence D is an eternal independent set. Hence α ∞ (G) ≥ |D| = m i (G).
There exist graphs for which equality in the chain give in Theorem 2.1 does not necessarily hold. Consider C 5 , where 2 = α ∞ (C 5 ) > m i (C 5 ) = 1. There are also graphs for which α ∞ m (G) < m(G) such as K 3 with a pendant vertex attached to one of the vertices or a K 5 with a pendant vertex attached to one of the vertices.
It seems that a graph with large matching and with low chromatic number should force a large free matching and hence a large total-eternal independence number. We detail this relationship in the next proposition.
Proof. Let M be a maximum matching of cardinality m. The subgraph of G induced by M , denoted G * , has χ(G * ) = t ≤ k. Let A 1 , . . . , A t be the color classes of G * . Now the m edges of M are divided into t(t − 1)/2 pairs (A i , A j ). Hence, by averaging, for some pair (i, j), the pair (A i , A j ) contains at least m/(t(t − 1)/2 ≥ m/(k(k − 1)/2 = 2m/k(k − 1) edges from M and these edges form a free matching.

Clique Coverings
Observe that α ∞ (G) ≤ θ(G), for all graphs G, since α(G) ≤ θ(G), for all G (since no clique in a clique cover can contain more than one vertex from any independent set). Proposition 3.1 Let G be a connected triangle-free graph with θ(G) ≥ 2 and no isolated vertices. Then α ∞ (G) < θ(G).
Proof. Suppose to the contrary that α ∞ (G) = θ(G). Let C = C 1 , C 2 , . . . , C θ be a minimum clique cover. Note that |C i | ≤ 2 since G is triangle-free. Then an eternal independent set D must contain exactly one vertex from each C i . If we request a vertex v ∈ C i , v ∈ D, that vertex must switch to another vertex in C i (since every C j , j = i contains another vertex in D). Thus each C i must be a K 2 . Let u, v be two vertices of minimum distance in D and such that the cliques from C in which they are contained are connected by an edge. Clearly 2 ≤ dist(u, v) ≤ 3. If dist(u, v) = 2, then a request to one of them (which one depends on their locations) will switch one of them so that u and v are adjacent. If dist(u, v) = 3, then consecutive requests to both u and v will cause two switches resulting in u and v being adjacent.
Proposition 3.1 is sharp for infinitely many graphs. Let G consist of n paths of length three having a common vertex w, i.e., a star K 1,n where each edge is subdivided once. G is K 3 free with θ(G) = n + 1 and α ∞ (G) = n (because of the induced matching, see Theorem 4.1 below). We leave open the problem of characterizing the triangle-free graphs for which α ∞ (G) = θ(G) − 1.
As another example, for a triangle-free graph G on n vertices, cor(G) (which is again triangle-free) has the following properties: θ(cor(G)) = n > α ∞ (cor(G)) ≥ c √ n log n. The left hand-side come from Proposition 3.1, while the right side come from the Ramsey number R(K 3 , K n ) and the fact that α ∞ (cor(G)) = α(G), since it is well-known that if G is trianglefree, it has an independent set of cardinality at least c √ n log n and this is sharp.
We can ask for which connected graphs is α ∞ (G) = α(G) = θ(G)? It seems difficult to structurally describe these graphs but some observations are in order. If θ = 1, then α ∞ (G) = α(G) = θ(G). Now let θ(G) > 1 and C = {C 1 , C 2 , . . . , C k } be a minimum clique covering. Supposing α(G) = θ(G), we get that there is an independent set consisting of one vertex from each C i . In order for α ∞ (G) = α(G), clearly each C i must contain at least two vertices and no two C i 's that are both K 2 's can be joined by an edge. This leads us to the following.
, then by the pigeonhole principle there must simultaneously exist two guards within the same clique from some minimum clique-covering. But two such guards cannot be on independent vertices. Thus α ∞ (G) = θ(G).
For the other direction, let us assume α ∞ (G) = θ(G). From the observation above, we may assume that θ(G) > 1.
Using the notation from above, each clique C i from clique cover C can contain at most one edge from any matching. Further, each C i is a clique with at least two vertices, because if any C i is a K 1 , then we can easily force a switch that destroys independence. Any eternal independent set D of cardinality θ(G) contains exactly one vertex from each clique of clique cover C, since α ∞ (G) = α(G) = θ(G). Denote the vertices in D as D = {v 1 , v 2 , . . . , v k }, k = θ(G) and let v i ∈ C i . Obviously D is an independent set. If C i is a K i , i > 1, let . .} be the other vertices in the clique C i along with v i . For simplicity in what follows, we shall omit the superscript on a u k i vertex when it is clear from the context and refer to these as u i type vertices.
We construct a modified graph G as follows. If any u i type vertex is adjacent to any v j , j = i, delete that u i vertex (since if v i were attacked first in G, the guard could not switch to that u i vertex without destroying independence). Then if any K 2 's in the resulting graph have a u i type vertex adjacent to any u j type vertices, j = i, delete all such u j vertices (since such vertices cannot be switched to without destroying independence). In the resulting graph G , what must remain are K 2 components and other cliques with more than two vertices (any two such cliques with more than two vertices may be connected via a limited number of edges). If there are any K 1 components in G , then D is not an eternal independent set, since we could force a switch in G that destroys independence. The K 2 components can be removed and placed into the induced matching, M , that we are building. So only cliques with more than two vertices remain in the reduced graph G . Observe that neither (u i v j ) nor (u j v i ) are edges for any distinct cliques C i , C j , in the clique cover C when restricted to G .
Let D ⊆ D be the vertices of D that are in G . Let D = {v 1 , v 2 , . . . , v t } and let D * = D \ D . Considering G, start with guards on the vertices of D and attack all the vertices of D * and then attack each of the vertices in D , with v i ∈ D switching to a vertex u a i , for some a. The set of u a i vertices are an independent set. Either the edges switched across form an induced matching or some u a i is adjacent to some v b . But there are no such adjacencies in the graph G . Hence we can add these edges switched across to the K 2 components above to form an induced matching of G.
It seems interesting to find graphs classes for which α ∞ m (G) = θ(G); C 4 and P 4 are two examples where equality holds, but which have α ∞ m (G) > m i (G). Proof. If there is a maximum induced matching with t edges, then a vertex can be switched along each of these edges eternally; therefore there exists an eternal independent set with t vertices.

Bipartite Graphs
Suppose there exists an eternal independent set D with k vertices. We can request a set vertices be attacked such that all these vertices are in A, say on a 1 , a 2 , . . . , a k . Because if some vertex v ∈ D is not in A (i.e., v / ∈ A), then it must be that V ∈ B; so therefore we can attack v. Thus the guard on v cannot stay in B and must move to A , so we can repeat requesting until all vertices are in A, say a 1 , . . . , a k , |D| = k.
Let b i be the vertex in B such that if a i is attacked next, a i is switched to b i . Then b i cannot be adjacent to any a k , k = i, otherwise independence would be destroyed, contradicting that the set of vertices was an eternal independent set.
So consider the set of edges e i = (a i , b i ) which is a matching. There are no edges between the a i being all in A , there are no edges between the b i being all in B. If there is an edge between e i and e j (j = i) then it is either (a i , b j ) or (a j , b i ) which is impossible. As all b i are independent from all a j (j = i), this is an induced matching.
This property does not hold for all non-bipartite graphs: C 5 is an example of a graph with α ∞ (C 5 ) = 2 and m i (C 5 ) = 1. Furthermore, observe that any tree T with θ(T ) = α(T ) > 2 has θ(T ) > m i (T ). This is because in order for θ(T ) = m i (T ), T would have to all the edges in the tree in some minimum clique covering. But two edges that are joined by an edge cannot be in the same induced matching.
A linear-time algorithm for finding a maximum induced matching in a tree is given in [10,11]. Thus, using Theorem 4.1, one can find give an algorithm that computes the order of the maximum-eternal independent set in a tree in linear time. We give here an alternative linear time algorithm which is simpler and directly finds a maximum eternal independent set in a tree.
A stem in a tree is a vertex adjacent to a leaf and the height of a tree with specified root vertex r is the maximum distance from r to any leaf.
If the height of a tree T with at least two vertices is one, then the maximum eternal independent set is of size 1. Otherwise, suppose the height of tree T is more than one. In this case, we find a root vertex r of T that is not a stem, which necessarily exists as T is not a K 1,m for any m ≥ 1. The root may be a leaf.
We shall build a set D that will eventually contain the vertices of a maximum eternal independent set. Pick a stem v 1 of maximum distance from r. Let w be parent of v 1 (possibly, w = r). Let v 1 , . . . , v k be all the stems that are children of w. Place each v i in D. Remove all children and grandchildren of w from T , letting the resulting tree be T . Proceed recursively on T , terminating when the tree T has height at most one. If T has height at most one, then no more vertices will be added to D.
We now prove the algorithm finds a maximum eternal independent set.
Proof. When the height of T is one, α ∞ (T ) = m i (T ) = 1.
Now assume the height of T is h > 1. When h = 2, D consists of the children of r. In this case |D| = m i (G). Let us suppose h > 2. Consider the tree T as described in the algorithm. Clearly the maximum eternal independent set of T − T consists of k vertices: v 1 , . . . , v k (none of which are leaves), since a guard on v i can move to one of its children. Note that w is a leaf in T . Then the eternal independent set D found by the algorithm in T is a largest eternal independent set not containing w. Therefore, D ∪ {v 1 , . . . , v k } is a maximum eternal independent set of T , since a guard in v i can move to its child when attacked.
In other words, D consists of vertices labeled v i at any time in the algorithm. These vertices form an independent set; therefore no vertex ever labeled w can be part of this same independent set. No vertex labeled w can be subsequently labeled as v i , as w becomes a leaf in the tree T . If we think of the edges that guards move across in this scheme as a matching, then D consists of one endvertex from each edge in this matching. Each neighbor of a w vertex is an endvertex of an edge in this matching.
Furthermore, the root, r, cannot be part of this independent set unless it is labeled as v 1 at some point in the algorithm, otherwise a guard on r would have to move to one of its children when r is attacked, but this child was once a w vertex (and thus is adjacent to some v i vertex that may have a guard on it). Proof. Recall that in the m-eternal independent set problem, we may move as many guards as needed (including the possibility of a total-switch), as long as we move the guard from the attacked vertex.
Let M = {e i = (v i , u i ) : i = 1, . . . , k} be a matching of maximum cardinality, so k = m(G). Since we can do total-switches from the endvertices of M in A to the endvertices of 6. Describe some graph classes for which α ∞ m (G) = α(G). Well-covered graphs (i.e., graphs in which all maximal independent sets have the same cardinality) have this property, since there exists a perfect matching between the vertices in the symmetric difference of any two maximal independent sets, c.f. [4]. When a vertex is attacked, there exists a maximal independent set containing a neighbor of the attacked vertex (since each vertex belongs to some maximal independent set) and there exists a perfect matching that can be switched across between the vertices in the symmetric difference of these two maximal independent sets.
Cayley graphs are another class of graphs that may have this property.