Transitivity on Minimum Dominating Sets of Paths and Cycles

Transitivity on graphs is a concept widely investigated. This suggest to analyze the action of automorphisms on other sets. In this paper, we study the action on the family of γ-sets (minimum dominating sets), the graph is called γ-transitive if given two γ-sets there exists an automorphism which maps one onto the other. We deal with two families: paths Pn and cycles Cn. Their γ-sets are fully characterized and the action of the automorphism group on the family of γ-sets is fully analyzed.


Introduction
In graph theory there are several families of graphs that have been studied from different points of view, for example, in [1], infinite sequences of graphs determined by its generalized spectrum from certain small starter graphs are constructed. In [2], the chromatic number of sunlet and bistar families is found and in [3], the transversal domination number for some families, such as: bipartites, multipartites, paths, cycles, wheels and sunlets, is computed.
Domination in graphs and its variations are topics comprehensively studied and many papers have been written about it, since γ-sets (dominating sets of minimum cardinality) or the domination number is an NP-complete problem for arbitrary graphs [4], most of them focus on finding and describing bounds for the domination number, for example in terms of the order [5], minimum or maximum degree [6] or size [7], to name a few. The study of dominant sets, γ-sets and the domination number has attracted the attention of mathematicians due to the applications these concepts have in areas as diverse as social networks, chemical compounds, optimization problems, among others [8]. Moreover, there have been new treatments developed around these concepts, for example k-domination and distance k-dominating sets, see [9,10].
In this paper, we deal with paths and cycles, emphasizing on characterization of minimum dominating sets and the action of the automorphism group on these sets. In addition to being important by themselves, because of their graph-theoretic properties, these families belong to a widely studied class of graphs called "chemical graphs", broadly speaking they are graphs associated to a molecular structure of chemical compounds. For example, the molecular structure of butane (C 4 H 10 ) can be identified with the path P 4 and that of the cyclopentane (C 5 H 10 ) with the cycle C 5 .
By graph, we mean a simple graph with no loops and it is denoted by Γ = (V, E), where V and E are the vertices and edges sets, respectively. We use the following concepts, notation and results of graph theory. Let Γ = (V, E) be a graph: the domination number of Γ is: is a dominating set of cardinality γ; that is, a minimum dominating set; for Γ connected and u, v ∈ V, the distance between u and v is: An automorphism of Γ is an element σ ∈ S V ; that is, a permutation of V, which satisfies we denote the set of all automorphisms of Γ by Aut Γ, which is, in fact, a subgroup of S V . In what follows, S n denotes the symmetric group on n letters. We use freely the following facts about the automorphism groups we are interested in: • Aut P n ∼ = S 2 , with P n the path graph; • Aut C n ∼ = D n , with C n the cycle graph and D n the dihedral group (the rigid motions taking a regular n-gon back to itself, with the composition) which has the presentation ρ, µ : ρ n = 1 = µ 2 , ρµ = µρ −1 .
An action of a group G on a set X is a function A : G × X → X such that
A(1, x) = x, for all x ∈ X, where 1 ∈ G is the identity.
If x ∈ X, the orbit of x is the set and the collection of all orbits give a partition of X. The action is called transitive if there is only one orbit; that is, given It is clear that any subgroup G < S X acts on X in a natural way as follows

Results
It is known that if Γ = (V, E) then Aut Γ acts naturally on V and on E, when this action is transitive on V, Γ is called vertex-transitive, and edge-transitive if it is on E. In [11], γ-transitive graphs are defined, they are those where the action of Aut Γ on minimum dominating sets is transitive; that is, given D 1 , D 2 ∈ D γ , there is σ ∈ Aut Γ such that σ(D 1 ) = D 2 . We may observe easily the following facts: • any graph with just one minimum dominating set is trivially γ-transitive; • if the cardinality of D γ is greater than the order of Aut Γ, then Γ is not γ-transitive.
In this section we fully characterize γ-sets and analyze in detail the action of Aut Γ on D γ for path and cycle graphs to decide if they are γ-transitive. If they are not, then we compute all the orbits.

Path Graphs
Let n be a positive integer and consider the path graph P n = (V, E) which is defined as follows:  3 show the P 21 , P 20 and P 19 graphs, respectively. We may observe that deg v i = 1, for i = 1, n; 2, for i = 2, . . . , n − 1.
A well known fact is that γ(P n ) = k, for n = 3k, 3k − 1 and 3k − 2.The following lemmas characterize γ-sets for these graphs, they are properly counted and the action of the automorphism group on D γ is analyzed.
thus, D is not a dominating set. Conversely, if deg x i = 2, for i = 1, . . . , k and no pair of vertices share neighbors, we have Observe that Lemma 1 implies that is the unique γ-set for P 3k . Figure 1 shows this set for P 21 .
Lemma 2. For n = 3k − 1, D = {x 1 , . . . , x k } is a minimum dominating set if and only if one of the following conditions hold: there exists a unique pair of vertices x r , Proof. Suppose that D is a dominating set and let t denote the number of unordered pairs of distinct vertices x, y ∈ D such that N(x) ∩ N(y) = ∅; that is, |N(x) ∩ N(y)| = 1. We proceed by cases.
which contradicts D being dominating.

•
If D contains just one vertex of degree 1, then Finally, suppose that all vertices in D have degree 2, then Conversely, if N(x i ) ∩ N(x j ) = ∅, for i = j, and every vertex but one has degree 2, then and D is a dominating set. Now suppose that there is exactly one pair of vertices . . , k, as before, we obtain |N[D]| = 3k − 1 and, therefore, D is dominating.

Remark 1.
By Lemma 2 we may deduce that there are exactly two minimum dominating sets satisfying the first condition In fact, if D ⊆ V is a γ-set and we consider there is a one-to-one correspondence between these γ-sets and triplets (a, b, c) such that a + b + c = k, a, b > 0 and c = 0. Indeed, we may observe that the correspondence is given by there are exactly two distinct vertices x r , x s ∈ D sharing a neighbor, N[x i ] ∩ N[x j ] = ∅, for i, j = r, s, and every vertex in D has degree 2, but one; iii. deg x = 2 for all x ∈ D and there are exactly two distinct vertices x r , x s ∈ D such that x r x s ∈ E and Proof. Suppose that D is a dominating set and, as in the proof of Lemma 2, let t denote the number of unordered pairs of distinct vertices x, y ∈ D which share neighbors.
• Suppose that D contains vertices of degree 1.
If v 1 , v n ∈ D, since D is dominating and If D contains just one vertex of degree one, then If t = 0 then |N(D)| = 2k and, since |N[D]| = 3k − 2, necessarily |D ∩ N(D)| = 2 and the two vertices in this intersection form an edge, moreover, no other vertices share neighbors, thus, D satisfies (iii). If t ≥ 1, we get |N(D)| = 2k − t, obtaining t = 2 and D satisfies (iv).
Conversely, if D satisfies (i), then If D satisfies condition (ii), we get Analogously the other cases. Therefore, in any case, D is a dominating set.
It is important to note that condition (iv) is true for k ≥ 3. Considering the triplets (a, b, c) of nonnegative integers such that a + b + c = k, as above, this lemma implies the following facts.

Proposition 1.
There is a one-to-one correspondence between minimum dominating sets that satisfies condition (ii) and triplets (a, b, c) such that a, b > 0 and c = 0 or b, c > 0 and a = 0. Thus, the number of γ-sets of this kind is 2(k − 1).
Proof. If D satisfies (ii), it contains a vertex of degree 1, say v 1 , since there are exactly two vertices sharing a neighbor they must be of the form v r and v r+2 ; moreover, preceding vertices to v r have disjoint closed neighborhoods, so that the first vertices should be v 1 , v 4 , . . . , v 3b−2 = v r and, by the same argument, the vertices after Conversely, given a triplet (a, b, c) such that a + b + c = k and a = 0 or c = 0, we may construct easily the corresponding set as follows Therefore, there are 2(k − 1) γ-sets satisfying condition (ii).

Proposition 2.
There is a one-to-one correspondence between minimum dominating sets that satisfy condition (iii) and triplets of natural numbers (a, b, c) such that b = 0 and a, c > 0. Thus, the number of γ-sets corresponding to these triplets is k − 1.
Proof. Let D be a γ-set satisfying condition (iii), thus every vertex has degree 2. It contains two vertices forming an edge, say v r , v r+1 , and N[x i ] ∩ N[x j ] = ∅, for i, j = r, r + 1. Since every vertex has degree 2 and preceding vertices to v r have disjoint closed neighborhoods, the first vertices should be v 2 , v 5 , . . . , v 3a−1 = v r , which implies v r+1 = v 3a and the same argument shows that the following vertices are v 3(a+1) , v 3(a+2) , . . . , v 3(k−1) . Thus, the triplet corresponding to D is (a, 0, k − a). Conversely, given a triplet (a, 0, c) such that a + c = k and a, c > 0 we may construct the γ-set which clearly satisfies condition (iii). Therefore, there are k − 1 γ-sets satisfying this condition.

Proposition 3.
There is a one-to-one correspondence between minimum dominating sets that satisfy condition (iv) and triplets of natural numbers (a, b, c) such that a, b, c > 0. Moreover, the number of γ-sets satisfying this condition is Proof. Let D be a γ-set satisfying condition (iv); that is, every vertex has degree 2, it contains four vertices (not necessarily different) sharing a neighbor, say v r , v r+2 , v s , v s+2 , and N[x i ] ∩ N[x j ] = ∅, for i or j = r, r + 2, s, s + 2. Applying an analogous argument as in the previous cases, we may infer that the first vertices are v 2 , v 5 , . . . , v 3a−1 = v r , the vertices after v r+2 = v 3a+1 must be v 3(a+1)+1 , v 3(a+2)+1 , . . . , v 3(a+b)+1 = v s and the vertices following v s+2 = v 3(a+b+1) are v 3(a+b+2) , v 3(a+b+3) , . . . , v 3(k−1) . Thus, the triplet corresponding to D is (a, b, k − (a + b)). Conversely, given a triplet (a, b, c) such that a + b + c = k and a, b, c > 0, we get the following γ-set which fulfills condition (iv).
Recall that Aut P n = {1, σ} ∼ = S 2 , for n ≥ 2, where 1 is the identity and σ is the automorphism given (on subindices) by i −→ n + 1 − i, and note that Theorem 1. Let n be a positive integer, considering the action of Aut P n on D γ , we have: for n = 3k − 1 there are k−1 2 + 1 orbits; iii. for n = 3k − 2 and k ≥ 3, there are k−2 Proof. By the comment made after Lemma 1, we know that there is just one γ-set for P 3k , so that P 3k is trivially γ-transitive. If n = 3k − 1, by Lemma 2 we know that there are γ-sets of two kinds: those which contain a vertex of degree 1, denoted by D 1 and D 2 , and those which are in correspondence with triplets (a, k − a, 0); moreover, σ(i) ≡ 2i mod 3. Clearly, σ(D 1 ) = D 2 and the image under σ of the γ-set corresponding to (a, k − a, 0) is the one corresponding to (k − a, a, 0); that is, Thus, there are 1 + k−1 2 orbits. If n = 3k − 2, Lemma 3 implies that there are four kinds of γ-sets and σ(i) ≡ 2i + 2 mod 3. We proceed by cases.
In the second case, γ-sets are in correspondence with triplets (a, b, c) such that a, b > 0 and c = 0 or b, c > 0 and a = 0. If D corresponds to (0, b, c), then σ(D) corresponds to (c, b, 0), and viceversa. A γ-set corresponding to (a, b, 0) is mapped to that corresponding to (0, b, a). This implies that the orbit of one of these sets contains exactly two of them. Thus, there are k − 1 orbits.
In the third one, the correspondence is with triplets (a, b, c) such that b = 0 and a, c > 0. The image of the set corresponding to (a, 0, c) is the one that corresponds to (c, 0, a). Note that the orbit is a singleton if and only if a = c, which happens when k is even and 2a = k, so there are k−1 2 orbits. Finally, in the last case the correspondence is with triplets (a, b, c) such that a, b, c > 0. We may observe that if D corresponds to (a, b, c), then σ(D) corresponds to (c, b, a), and, as in the last case, the orbit is a singleton if and only if a = c. In this sub-case the triplets are (a, k − 2a, a) and, since k − 2a > 0, there are k−2 2 of these sets. Thus, there are 1 orbits with two elements.
The remaining cases n = 3k − 2 for k = 1, 2 are as follows: P 1 is trivially γ-transitive and we may observe easily that P 4 has four γ-sets and three orbits

Cycle Graphs
For an integer n ≥ 3, consider the cycle graph C n = (V, E), where V = {v 1 , . . . , v n } and E = {v 1 v 2 , . . . , v n−1 v n , v n v 1 }, Figures 4-6 show the C 21 , C 20 and C 19 graphs, respectively. In this section, we use freely that the domination number of C n is k, for n = 3k, 3k − 1 and 3k − 2.  The proof of the following lemma is completely analogous to that of Lemma 1 and it is omitted. This lemma implies that there are exactly three γ-sets for C 3k , namely, thus, D is a γ-set.
By Lemma 5, we may observe that there is a one-to-one correspondence between D γ and V given by D i ←→ v i , with v i the unique neighbor shared by two vertices of D i , thus, there are n sets of this kind: where subindices are taken modulo n = 3k − 1. In Figure 5 is shown the set D 1 for C 20 . Conversely, if D = {x 1 , . . . , x k } is a set satisfying condition (i), we get which implies that D is a γ-set. Analogously if D satisfies condition (ii).
There is a one-to-one correspondence between γ-sets satisfying the first condition and E. This correspondence is given by Figure 6a shows the set D 1 corresponding to e 1 for C 19 .

Proposition 4.
For those γ-sets that satisfy (ii), we have two cases: • x b = x r , obtaining n γ-sets of this kind; or • x r , x s , x a and x b are all different, getting k−2 2 n γ-sets of this kind for k odd and n+k−2 2 for k even.
Proof. In the first case, there is a one-to-one correspondence between these γ-sets and V as follows D i ←→ x r , where x r is the vertex that shares a neighbor with other two, thus, there are n = 3k − 2 sets of this kind.
In the second one, we may obtain a one-to-one correspondence between these γ-sets and pairs of vertices u, v such that d(u, v) = 3l + 2, for l = 1, 2, . . . , k−2 2 , namely, Note that if 3l + 2 < n 2 , there are exactly n pairs of vertices verifying this condition: v i and v i+3l+2 , for 1 = 1, . . . , n and i + 3l + 2 taken modulo n. When 3l + 2 = n 2 , which happens if and only if k = 2d and l = d − 1, there are just n 2 pairs of these vertices: v i and v i+3d−1 , for i = 1, . . . , 3d − 1. Thus, there are k−2 2 n γ-sets of this kind for k odd and n+k−2 2 for k even. Figure 6b shows the set D 1 , corresponding to v 1 , and in Figure 6c is shown the set D 1,9 , corresponding to the pair {v 1 , v 9 }, for C 19 .
It is known that Aut C n ∼ = D n = ρ, µ : ordρ = n, ordµ = 2, µρ = µρ −1 , where ρ may be considered as a rotation by 360 • /n and µ the reflection across the symmetry line that fixes v 1 of the regular n-gon, thus, Theorem 2. Let n be a positive integer, considering the action of Aut C n on D γ , we have: i. C 3k is γ-transitive; ii.
For n = 3k − 2, Lemma 6 implies that there are two kinds of γ-sets by cases.
In the second case, there are two subcases. In the first one, γ-sets are in correspondence with the vertex set and clearly the orbit has n elements. In the second, the correspondence is between γ-sets and unordered pairs of vertices whose distance is 3l + 2, for l = 1, 2, . . . , k−2 2 . Let D i be the γ-set corresponding to the pair of vertices v i , v i+3l+2 , thus ρ j−1 (D 1 ) = D j and µ(D i ) = D n−(i+3l) . Therefore, O(D i ) comprises all γ-sets of this kind, n if 3l + 2 < n 2 and n 2 if 3l + 2 = n 2 .

Concluding Remarks
In this work we have studied γ-transitivity for paths and cycles observing that our main tool is to characterize completely minimum dominating sets for each family and when a graph is not γ-transitive the orbits are computed. We have already observed that when the characterization has at least two kinds of γ-sets, the graph does not satisfy the property and has as many orbits as these kinds. As we noted above, this work was motivated by a previous research on Platonic graphs [11], in which is shown which of them are γ-transitive and why some are not. The main difference with this is that, when a graph does not have this feature, the orbits are exhibited.
There are several questions to be answered; for example, what families happen to have this attribute or are there any necessary or sufficient conditions for ensuring this property is fulfilled?