Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of the Layer Sun Graph

. Let G be a ﬁnite, connected graph of order of, at least, 2 with vertex set V ( G ) and edge set E ( G ) . A set S of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of S . The minimal doubly resolving set of vertices of graph G is a doubly resolving set with minimum cardinality and is denoted by ψ ( G ) . In this paper, ﬁrst, we construct a class of graphs of order 2 n + Σ k − 2 r � 1 nm r , denoted by LSG ( n, m, k ) , and call these graphs as the layer Sun graphs with parameters n , m , and k . Moreover, we compute minimal doubly resolving sets and the strong metric dimension of the layer Sun graph LSG ( n, m,k ) and the line graph of the layer Sun graph LSG ( n, m, k ) .


Introduction
In this paper, suppose G is a finite, simple connected graph of order of, at least, 2, with vertex set V(G) and edge set E(G). If x and y are vertices in the graph G, then the distance x from y in G is denoted by d G (x, y) or simply d(x, y), where d(x, y) is the length of the shortest path from x to y. e line graph of a graph G is denoted by L(G), with vertex set V(L(G)) � E(G) and where two edges of G are adjacent in L(G) if and only if they are incident in G, see [1]. Vertices x, y of the graph G are said to doubly resolve vertices u, v of (v, y). A set S of vertices of the graph G is a doubly resolving set of G if every two distinct vertices of G are doubly resolved by some two vertices of S. e minimal doubly resolving set of vertices of graph G is a doubly resolving set with minimum cardinality and is denoted by ψ(G). e notion of a doubly resolving set of vertices of the graph G was introduced by Cáceres et al. [2]. A vertex w strongly resolves two vertices u and v if u belongs to a shortest v − w path or v belongs to a shortest u − w path. A vertex set S of the graph G is a strong resolving set of G if every two distinct vertices of G are strongly resolved by some vertex of S. A strong metric basis of G is denoted by sdim(G) defined as the minimum cardinality of a strong resolving set of G. e notion of a strong metric dimension problem set of vertices of the graph G was introduced by A. Sebö and E. Tannier [3] and further investigated by O. R. Oellermann and Peters-Fransen [4]. e minimal doubly resolving sets for jellyfish and cocktail party graphs have been obtained in [5]. For more results related to these concepts, see [6][7][8][9][10][11][12][13][14][15][16]. In this paper, first, we construct a class of graphs of order 2n + Σ k−2 r�1 nm r , denoted by LSG(n, m, k), and call these graphs as the layer Sun graphs with parameters n, m, and k, which is defined as follows: Let n, m, k be integers such that n, k ≥ 3, m ≥ 2 and G be a graph with vertex set V(G) � V 1 ⋃ V 2 ⋃ . . . ⋃ V k , where V 1 , V 2 , . . . V k are called the layers of G such that . . , v n , and for l ≥ 3, we have V l � B (l) where K m is the complement of the complete graph on m vertices. Now, suppose that every vertex i in the cycle C n is adjacent to exactly one vertex in the layer V 2 , say v i ∈ V 2 , and every vertex v i in the layer V 2 is adjacent to exactly m vertices , and then, the resulting graph is called the layer Sun graph LSG(n, m, k) with parameters n, m, and k. Also, for l ≥ 3, we recall that B (l) i j as the components of the layer V l , 1 ≤ i ≤ n, 1 ≤ j ≤ m l− 3 . In particular, we say that two components B (l) It is natural to consider its vertex set of the layer Sun graph LSG(n, m, k) as partitioned into k layers. e layers V 1 and V 2 consist of the vertices 1, 2, . . . , n { } and v 1 , v 2 , . . . , v n , respectively. In particular, each layer V l (l ≥ 3) consists of the nm l− 2 vertices. Note that, for each vertex i in the layer V 1 and every vertex In this paper, we consider the problem of determining the cardinality ψ(LSG(n, m, k)) of minimal doubly resolving sets of the layer Sun graph LSG(n, m, k). First, we find the metric dimension of the layer Sun graph LSG(n, m, k); in fact, we prove that if n, k ≥ 3 and m ≥ 2, then the metric dimension of the layer Sun graph LSG(n, m, k) is nm k− 2 − nm k− 3 . Moreover, we consider the problem of determining the cardinality ψ(LSG(n, m, k)) of minimal doubly resolving sets of LSG(n, m, k) and the strong metric dimension for the layer Sun graph LSG(n, m, k) and the line graph of the layer Sun graph LSG(n, m, k). e graph LSG (3,3,4) is shown in Figure 1.

Definitions and Preliminaries
For an ordered subset W � w 1 , w 2 , . . . , w k of vertices in the graph G and a vertex v of G, the metric representation of v with respect to W is the k-vector . If every pair of distinct vertices of G has different metric representations, then the ordered set W is called a resolving set of G. If the set W is as small as possible, then it is called a metric basis of the graph G. We recall that the metric dimension of G, denoted by β(G), is defined as the minimum cardinality of a resolving set for G. Proposition 1. Let G be a graph. It is well known that a doubly resolving set is also a resolving set and β(G) ≤ ψ(G). In particular, every strong resolving set is a resolving set and β(G) ≤ sdim(G).

Minimal Doubly Resolving Sets and the Strong Metric
Dimension for the Layer Sun Graph LSG(n, m, k) Theorem 1. Let G � LSG(n, m, k) be the layer Sun graph which is defined already. Suppose that n, m, k are integers such that n, k ≥ 3 and m ≥ 2. en, the metric dimension of . . , V k are called the layers of vertices in the layer Sun graph LSG(n, m, k), which is defined already. It is clear that if W is an ordered subset of the layers V 1 ⋃ V 2 ⋃ . . . ⋃ V k−1 , then W is not a resolving set in LSG(n, m, k). We may assume that the layer V k is equal to where In the following cases, it can be shown that the metric dimension of the layer Sun graph LSG(n, m, k) is nm k− 2 − nm k− 3 .
Case 1: let W be an ordered subset of the layer V k in the layer Sun graph LSG(n, m, k) such that (2) Hence, We know that the cardinality of W is us, W is not a resolving set in LSG(n, m, k) Case 2: let W be an ordered subset of the layer V k in the layer Sun graph LSG(n, m, k) such that Hence,

Complexity
We know that |W| � nm k− 2 − 2. erefore, the metric representation of two vertices us, W is not a resolving set in LSG(n, m, k). Case 3: let W be an ordered subset of the layer V k in the layer Sun graph LSG(n, m, k) such that Hence, We know that |W| � nm k− 2 − nm k− 3 . We can show that all the vertices in V(G) − W have different representations with respect to W. Let u be the vertex of the layer V 1 � V(C n ) � 1, 2, . . . , n { }. We can assume without loss of generality that We can assume without loss of generality that In a similar way, we can show that all the vertices in the layers V 3 , . . . , V k−1 have different representations with respect to W. In particular, for every vertex u ∈ is implies that W is a resolving set in LSG(n, m, k). From the abovementioned cases, we can be concluded that the minimum possible cardinality of a resolving set in □ Theorem 2. Let G � LSG(n, m, k) be the layer Sun graph which is defined already. Suppose that n, m, k are integers such that n, k ≥ 3 and m ≥ 2. en, the cardinality of minimum doubly resolving set of the LSG(n, m, k) is nm k− 2 .
Proof. In the following cases, it can be shown that the cardinality of minimum doubly resolving set of the layer Sun graph LSG(n, m, k) is nm k−2 .
Case 2: now, let the subset of vertices in LSG(n, m, k) be In a similar fashion as in In a similar fashion as in eorem 1, we can show that all the vertices in the layers V 1 , V 2 , . . . , V k−1 of LSG(n, m, k) have different representations with respect to W. So, this subset is also a resolving set in LSG(n, m, k) of cardinality nm k− 2 . We show that this subset is a doubly resolving set for LSG(n, m, k). It is sufficient to prove that for two vertices u and v in LSG(n, m, k), there are vertices Consider two vertices u and v in LSG(n, m, k). en, we have the following.
Case 3.1: suppose that both vertices u and v lie in the layer V 1 . Hence, there are r, s ∈ 1, 2, . . . , n { } such that u � r and v � s. Moreover, we know that the layer Sun graph LSG(n, m, k) has the property that, for each vertex r in the layer V 1 , there is some vertex such as In the same way, there is some vertex such as y � (v s j , t) k in the component B (k) s j , in the layer V k , at distance k − 1 from v. In particular, it is easy to prove that d(u, x) − d(u, y) < 0 because d(u, y) ≥ k.  (v, y). Case 3.3: suppose that both vertices u and v lie in the layer V l , l ≥ 3 such that these vertices lie in the one component of the layer V l , say B (l) i j , 1 ≤ i ≤ n, 1 ≤ j ≤ m l− 3 . In this case, d(u, v) � 2. Moreover, we know that the layer Sun graph LSG(n, m, k) has the property that, for each vertex u ∈ B (l) i j in the layer V l , there is a component of the layer Case 3.4: suppose that both vertices u and v lie in the layer V l , l ≥ 3 such that these vertices lie in the two distinct components of the layer V l . We can assume without loss of generality that u ∈ B (l) and v ∈ B (l) q j 2 , 1 ≤ p, q ≤ n, and 1 ≤ j 1 , j 2 ≤ m l− 3 . Moreover, we know that the layer Sun graph LSG(n, m, k) has the property that, for each vertex u ∈ B (l) In the same way, for the vertex v ∈ B (l) In the following, let two components B (l) (v, y). Now, let two components B (l) y). Case 3.5: suppose that vertices u and v lie in distinct layers V a , V b , respectively. Note that if a � 1 and b � 2, a � 1 and b > 2, or a � 2 and b > 2, there is nothing to do. Now, let 3 ≤ a < b. Hence, there is a component of the layer V a , say B (a) z) and d(u, x) ≠ d(v, x). Note that if i ≠ p, then there is a component of the layer V k , say B (k) p s , 1 ≤ p ≤ n, 1 ≤ s ≤ m k− 3 such that for any vertex  LSG(n, m, k) is nm k− 2 − 1.
Proof. In the following cases, it can be seen that the cardinality of the minimum strong resolving set of the layer Sun graph LSG(n, m, k) is nm k− 2 − 1.
Case 1: we know that the ordered subset W of vertices in equation (11) in the layer V k of the layer Sun graph LSG(n, m, k) is a resolving set for LSG(n, m, k) of cardinality nm k− 2 − nm k− 3 . Now, let . . . , (v n m k−3 , 1) k }. By considering distinct vertices u, v ∈ N, we can show that there is not a vertex w ∈ W such that u belongs to a shortest v − w path or v belongs to a shortest u − w path because the valency of every vertex in the layer V k is one. So, this subset is not a strong resolving set for G. us, we can be conclude that if W is a strong resolving set for graph G, then |W| ≥ nm k− 2 − 1 because |N| must be less than 2.
Case 2: on the other hand, we can show that the subset W of vertices in equation (12) in the graph G is a resolving set for graph G. We show that this subset is a strong resolving set in graph G. It is sufficient to prove that every two distinct vertices u, v ∈ V(G) − W are strongly resolved by a vertex w ∈ W. en, we have the following: Case 2.1: suppose that both vertices u and v lie in the layer V 1 . Hence, there are r, s ∈ 1, 2, . . . , n { } such that u � r and v � s. Moreover, we know that the layer Sun graph LSG(n, m, k) has the property that, for each vertex r in the layer V 1 , there is a component B (k) r j , 1 ≤ j ≤ m k− 3 in the layer V k such that, for every vertex such as w ∈ B (k) r j , we have d(u, w) � k − 1 and d(v, w) > k − 1, and hence, u belongs to a shortest w − v path. Case 2.2: now suppose that both vertices u and v lie in the layer V 2 . In a similar way as in Case 2.1, we can show that the vertices u and v are strongly resolved by a vertex w ∈ W. Case 2.3: suppose that both vertices u and v lie in the layer V l , l ≥ 3 such that these vertices lie in the one component of the layer V l , say B (l) i j , 1 ≤ i ≤ n, 1 ≤ j ≤ m l− 3 . In this case, d(u, v) � 2. Moreover, we know that the layer Sun graph LSG(n, m, k) has the property that, for each vertex u ∈ B (l) i j in the layer V l , there is a component of the layer V k , say B (k) i r , 1 ≤ i ≤ n, 1 ≤ r ≤ m k− 3 such that for any vertex w ∈ B (k) i r , we have d(u, w) � k − l, and hence, u belongs to a shortest w − v path. Case 2.4: suppose that both vertices u and v lie in the layer V l , l ≥ 3 such that these vertices lie in the two distinct components of the layer V l . We can assume without loss of generality that u ∈ B (l) and v ∈ B (l) q j 2 , 1 ≤ p, q ≤ n, and 1 ≤ j 1 , j 2 ≤ m l− 3 . Moreover, we know that the layer Sun graph LSG(n, m, k) has the property that, for each vertex u ∈ B (l) p j 1 in the layer V l , there is a component of the layer V k , say B (k) p r , 1 ≤ r ≤ m k− 3 such that for any vertex w ∈ B (k) p r , we have d(u, w) � k − l, and hence, u belongs to a shortest w − v path. Case 2.5: suppose that vertices u and v lie in distinct layers V a , V b , respectively. Note that if a � 1 and b � 2, a � 1 and b > 2, or a � 2 and b > 2, there is nothing to do. Now, let 3 ≤ b < a. Hence, there is a component of the layer V a , say B (a) p q . In particular, there is a component of the layer V k , say B (k) i r , 1 ≤ i ≤ n, 1 ≤ r ≤ m k−3 such that, for any vertex w ∈ B (k) i r , we have d(u, w) � k − a, and hence, u belongs to a shortest w − v path.
in the layer V l , 1 ≤ r ≤ n and r j in the layer V k , r ≠ 1 and 1 ≤ j ≤ m k− 3 such that, for every vertex such as w in the component B (k) r j , we have d(w, v) ≥ k, d(w, u) ≥ 2k − 1, and v belongs to a shortest w − u path. Now, let i ≠ 1; indeed, d(u, v) ≥ k, and hence, there is a component B (k) i j in the layer V k , 1 ≤ j ≤ m k− 3 such that, for every vertices such as w in the components and v belongs to a shortest w − u path.
us, from the abovementioned cases, we can be concluded that the cardinality of minimum strong resolving set of the layer Sun graph LSG(n, m, k) is nm k− 2 − 1.
and let D (l) ij � ⋃ m t�1 (u ij , t) l such that every (u ij , t) l is a vertex in the layer U l and D (l) ij � K m in the layer U l , where K m is the complete graph on m vertices. Now, suppose that every vertex i ∈ 2, 3, . . . , n { } in the cycle C n or the layer U 1 is adjacent to exactly two vertices in the layer U 2 say u i , u i−1 ∈ U 2 . In particular, for the vertex 1 in the layer U 1 , we have 1 adjacent to exactly two vertices in the layer U 2 , say u 1 , u n ∈ U 2 . Also, every vertex u i in the layer U 2 is adjacent to exactly m vertices (u i1 , 1) 3 , (u i1 , 2) 3 , . . . , (u i1 , m) 3 ∈ D (3) i1 ∈ U 3 , in particular for l ≥ 3, every vertex (u ir , t) l ∈ D (l) i r ∈ U l is adjacent to exactly m vertices ⋃ m t�1 (u ij , t) l+1 ∈ D (l+1) i j ∈ U l+1 , and then, the resulting graph is isomorphic with the line graph of the layer Sun graph LSG(n, m, k) with parameters n, m, and k; in fact, L(G) � H. Note that simply we use refinement of the natural relabelling of the line graph of the graph LSG(n, m, k). Also, for l ≥ 3, we recall D (l) i j as the components of U l , 1 ≤ i ≤ n, 1 ≤ j ≤ m l− 3 . In particular, we say that two components D (l) i j , D (l) r s 1 ≤ i, r ≤ n, 1 ≤ j, s ≤ m l− 3 are fundamental if i � r and j ≠ s. It is natural to consider its vertex set of the line graph of the layer Sun graph LSG(n, m, k) is also as partitioned into k layers. e layers U 1 and U 2 consist of the vertices 1, 2, . . . , n { } and u 1 , u 2 , . . . , u n , respectively. In particular, each layer U l (l ≥ 3) consists of the nm l− 2 vertices. Note that, for each vertex i in the layer U 1 and every vertex x ∈ D (l) ij ∈ U l , l ≥ 3, 1 ≤ j ≤ m l− 3 , we have d(i, x) � l − 1. In this section, we consider the problem of determining the cardinality ψ(L(G)) of minimal doubly resolving sets of the line graph of the layer Sun graph LSG(n, m, k). We find the minimal doubly resolving set for the line graph of the layer Sun graph LSG(n, m, k), and in fact, we prove that if n, k ≥ 3 and m ≥ 2, then the minimal doubly resolving set of the line graph of the layer Sun graph LSG(n, m, k) is nm k− 2 − nm k− 3 . Figure 2 shows the line graph of the graph LSG (3,3,4). Note that simply we use refinement of the natural relabelling of the line graph of the graph LSG (3,3,4).

Theorem 4.
Let G � LSG (3,3,4) be the layer Sun graph which is defined already. Suppose that n, m, k are integers such that n, k ≥ 3 and m ≥ 2. en, the cardinality of minimum doubly resolving set in the line graph of the graph G is nm k− 2 − nm k− 3 .