Power Graph and Exchange Property for Resolving Sets

A formula for computing the metric dimension of a simple graph, having no singleton twin, is given. A sufficient condition for a simple graph to have the exchange property, for resolving sets, is found. Some families of power graphs of finite groups, having this exchange property, are identified. The metric dimension of the power graph of a dihedral group is also computed.


Introduction
Resolving sets and metric basis enjoys a lot of success due to its applications in computer science, medical sciences and chemistry. The concepts of metric dimension and resolving sets were initially drafted for the metric spaces in in [3] but did not receive much attention may be because of continuous nature of standard Euclidean spaces R n . These concepts were utilized for about twenty years later in [4]. Since then it has been artistically used in graphs, robotics, pharmacy, networking and in many other elds. Recently, the theory of metric dimension has been generalized for metric spaces and geometric spaces [5,6].
Let Γ be a nite, simple, and connected graph with vertex set V(Γ) and edge set E(Γ). The distance d Γ (u, v) between two vertices u, v ∈ V(Γ) is the length of a shortest path between them. Let W = {w , w , . . . , w k } be an ordered set of vertices of Γ, and let v ∈ V(Γ). Then, the representation of v with respect to W is the k-tuple (d Γ (v, w ), d Γ (v, w ), . . . , d Γ (v, w k )). Two vertices u, v ∈ V(Γ) are said to be resolved by W if they have di erent representations. A subset W of vertices is a resolving set (or locating set) if every vertex of Γ is uniquely identi ed by its distances from the vertices of W. Thus, in a resolving set, every vertex of Γ has distinct representation. A resolving set of minimum cardinality is called a basis for Γ. The cardinality of such a resolving set is called the metric dimension of Γ and is denoted by β(Γ) (see [7][8][9][10][11][12][13][14]). A resolving set is said to be minimal if it contains no resolving set as a proper subset. As an application, S. Khuller [15] considered the metric dimension and basis of a connected graph in robot navigation problems. In [16], authors computed metric dimension of ower graph and some families of convex polytopes. Chartrand [19].
Whenever W and W are any two minimal resolving sets for Γ and for every u ∈ W , there is a vertex v ∈ W such that (W \{u}) ∪ {v} is also a minimal resolving set. Then, resolving sets are said to have the exchange property in the graph Γ (for details, see [1]). All the graphs considered in this paper are nite, simple and connected. Also, all the groups considered are nite. Furthermore, the exchange property of a graph Γ always means the property for resolving sets.
The open neighborhood of a vertex u ∈ V(Γ), denoted by N(u), is the set The two vertices u and v in a graph Γ are called twins, The relation ≡ is an equivalence relation (see [20]). Also, d Γ (u , w) = d Γ (u , w) for u ≡ u , and for all w ∈ V(Γ)\{u , u }. Let u denote the twin-set of u with respect to the relation " ≡ ", and let U(Γ) = {u|u ∈ V(Γ)} be the set of all such twin-sets.
The following de nition is helpful in proving the main results of this paper.

Main results
In this section, we formulate our main results. Proofs of these results are given in the next sections. Our rst result gives a formula to compute the metric dimension of a graph without singleton twins.
Moreover, every minimal resolving set is a basis of the graph Γ.
Our second result provides a su cient condition for a graph to have the exchange property.

Theorem 2.2. A graph Γ without singleton twins has the exchange property.
The concept of a matroid is a generalization of the notion of linear independence. More precisely, a nite matroid M = (H, I) is a pair (H, I), where H is a nite set called the ground set and I is a family of subsets of H called independent sets with the following properties: a) the empty set is independent; b) every subset of an independent set is independent (this property is called hereditary property); and c) if A and B are two independent sets of I with |A| > |B|, then there exist x ∈ A \ B, such that B ∪ {x} is also independent (this is called augmentation property).
A maximal independent set is called a basis of the matroid M. We say, as de ned in [1], a set W of vertices in a graph Γ is resolving independent, denoted by resindependent, if for every v ∈ W, W −{v} is not a resolving set. With this de nition, a maximal res-independent set is a minimal resolving set. This de nition of independence de nes a hereditary system in the graph Γ. The question of whether the exchange property holds in Γ is equivalent to the question whether the hereditary system M Γ is a matroid (see [21] for further details).
As an application of Theorem 2.2 to matroid theory, the following corollary can be used to de ne a matroid on a nite ground set.

Corollary 2.3. The hereditary system M Γ is a matroid for a graph Γ without singleton twin and every minimal resolving set is a basis for the matriod.
A matroid M is called strongly base orderable if for any two bases B and B there is a bijection π : Theorem 2.4. The matriod M Γ is strongly base orderable.

Conjecture 2.5. [2] For every matroid M, its toric ideal is generated by quadratic binomials corresponding to symmetric exchanges.
It is proved in [22] that the White conjecture is true for every strongly orderable matroid. Therefore, the conjecture is true for M Γ .
Let G be a nite group. An undirected power graph P G associated to G, is a graph whose vertices are the elements of G, and there is an edge between two vertices x and y if either x m = y or y m = x, for some positive integer m. The power digraph of G is a digraph P G with the vertex set G, and there is an arc from vertex x to y if x m = y, for some positive integer m. The directed power graph of a group was introduced by Kelarev and Quinn [23]. The de nition was formulated so that it applied to semigroups as well. The power graphs of semigroups were rst considered in [24][25][26]. All of these papers used only the brief term 'power graph', even though they covered both directed and undirected power graphs. The investigation of graphs of this sort is very important, because they have serious applications and are related to automata theory (see [27,28] and the books [29,30], where applications are presented). It is also explained in the survey [31] that the de nition given in [23] covers all undirected graphs as well. Chakrabarty, Ghosh, and Sen [32] also studied undirected power graphs of semigroups. Recently, many interesting results on the power graphs of nite groups have been obtained (see [16,[33][34][35][36]). It is obvious that the power graph of a nite group is always connected. For other results and open questions on power graphs, we refer to the survey [31].
In our next theorem, the metric dimension of the power graph of the dihedral group D n of order n is computed. Theorem 2.6. β(P D n ) = β(P Zn ) + n − , where Zn is a cyclic group of order n.
In the following theorem, we identify some nite groups whose corresponding power graph de ne a matroid on the group. Theorem 2.7. Let G be a nite group and P G be the power graph associated to G. Then, M P G is a matroid if G is cyclic and |G| = k + for positive integers k.

Proofs . Exchange property
Every vector in a nite dimensional vector space is uniquely determined (written as a linear combination) by the elements of a basis of the vector space. A basis of a vector space has the exchange property. Similarly, each vertex of a nite graph can be uniquely identi ed by the vertices of a minimal resolving set. Therefore, resolving sets of a nite graph behave like bases in a nite dimensional vector space. Unlike a linear basis of a vector space, the minimal resolving sets do not always have the exchange property. Results about the exchange property for di erent graphs can be found in the literature. For example, the exchange property holds for resolving sets in trees; for n ≥ , the exchange property does not hold in wheels Wn [1].
for all x ∈ V(Γ) which means that u and v are twins, a contradiction. Consequently, exactly one representative from each twin-set stays outside W . Therefore, The cardinality of a minimal resolving set W is ≥ β(Γ). Now, W must have exactly β(Γ) = n i= m i − n vertices.
Otherwise, W contains an entire twin-set u of a vertex u and W \{u} is again resolving set, a contradiction. Therefore, every minimal resolving set is a basis.

Proof of Theorem 2.2:
Let W and W be two di erent minimal resolving sets in a graph Γ, and let u ∈ W . If u ∈ W , then obviously {W \{u }} ∪ {u } is a minimal resolving set. For u ∉ W there exists a vertex u ∉ W such that u ≡ u . Otherwise, W contains an entire twin-set and W is not minimal by Theorem 2.1, a contradiction. By Lemma 3.1, u ∈ W and every vertex in V(Γ)\{u , u } is at same distance from the vertices u and u . Therefore, the vertices which are resolved by u are also resolved by u and vice versa. Consequently, ({W \{u }) ∪ {u } is again a minimal resolving set.
Proof of Theorem 2.4: Let W and W are two bases. De ne a bijection π : W → W as follows The graph Γ is without singleton twins. Therefore, W ∪ π(U) \ U is a minimal resolving set (basis for the matroid M Γ ) for all U ⊂ W .

De nition 3.3. [37] For elements x and y in a group G, write R{x, y}
An involution is a non-identity element of order in a group G. A resolving involution, in a power graph P G of a group G, is an involution w which satis es that there exist two vertices x, y ∈ V(P G )\w with R{x, y} = {x, y, w}. Let W(P G ) denotes the set of all resolving involutions of P G .
x , x } be the cyclic group of order . Note that R{x, y} = {u, v, x } for u ∈ {x, x } and v ∈ {x , x }. Therefore, x is a resolving involution of P G .
Let Ψ denote the set of noncyclic groups G such that there exists an odd prime p such that the following conditions hold (see [37]): (C ) the prime divisors of |G| are and p; (C ) the subgroup of order p is unique; (C ) there is no element of order in G; and (C ) each involution of G is contained in a cyclic subgroup of order p.
In the original paper [37], for a nite group G, the notations |G|; |U(G)|; and |W(G)| were used for |V(P G )|; |U(P G )|; and |W(P G )| respectively. We give the following results in our notations.
Corollary 3.6. [37] Suppose that n = p r · · · p rt t , where p , . . . , p t are primes with p < · · · < p t , and r , . . . , r t are positive integers. Let Zn denotes the cyclic group of order n. Then Proof. The neighborhood in the graph P D n , of every involution w ∈ B is {e}. Therefore, w = B. If x, y ∈ V(P D n )\B, then there are two possibilities: 1) x = a s , y = e, ≤ s ≤ n − ; 2) x = a s , y = a s , ≤ s , s ≤ n − . In the above two cases, one can see that R{x, y} ≠ {x, y, w}. Therefore, w ∈ B is not a resolving involution.
Proof of Theorem 2.6: By part (ii) of Lemma 3.7, every resolving involution in P D n belongs to the subgraph P a , corresponding to the cyclic subgroup a , of D n . Therefore, W(P D n ) = W(P a ). In the subgraph P a , the identity e and the generator a are twins. However, e is the unique singleton twin in P D n . By part (i) of Lemma 3.7, all w ∈ B are in the same twin-set. Therefore, the set U(P D n ) is the disjoint union of U(P a ); the twin-set of e, and the twin-set of w, for w ∈ B. Consequently, |U(P D n )| = |U(P a )| + . A dihedral group D n does not satisfy the condition (C ); therefore, D n ∉ Ψ. Now, put |V(P D n )| = |V(P a )| + n; |U(P D n )| = |U(P a )| + ; and |W(P D n )| = |W(P a )| in the equation of part (ii) of Theorem 3.5 to complete the proof.
To compute the exact value of β(P D n ), one can use Theorem 2.6 and corollary 3.6. Proof of Theorem 2.7: Let G be a cyclic group of odd order and y is a generator of G. Then, there is no involution in the group G. Also, part (ii) of Proposition 3.2 implies that y = e, and e is not a singleton twin. Therefore, by Lemma 3.7, the graph P G has no singleton twin. Hence, the exchange property holds in P G by Theorem 2.2.
The following example shows that the converse of Theorem 2.2 and Theorem 2.7 is not true.

Conclusions
We give a new formula for computing the metric dimension of a simple graph without singleton twins. We also give su cient conditions for a graph to have the exchange property for resolving sets. Moreover, we deduce a new way to de ne a matroid on nite group. It is proved that the new matroid is strongly base orderable and hence satis es the conjecture of White [2]. We also compute the metric dimension of the power graphs of dihedral groups. We did not encounter a power graph of a nite group which does not have the exchange property. Therefore, the following question makes sense to be posed. It is worth mentioning that the authors of [38] have cited the pre-published version of the present article and have answered Question 4.1. They, in fact, give a necessary and su cient condition for resolving sets to have the exchange property in the power graphs of nite groups.