On the Geodesic Identification of Vertices in Convex Plane Graphs

Department of Information Technology, Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics, (e Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan Department of Basic Sciences, Balochistan University of Engineering and Technology Khuzdar, Khuzdar 89100, Pakistan Department of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Pakistan School of Mathematics, Southeast University, Nanjing 210096, China Yonsei Frontier Lab, Yonsei University, Seoul 03722, Republic of Korea


Introduction
e identification of vertices of a graph using various graph parameters is a fascinating problem for researchers. In the literature, almost over 1000 articles are contributed to explore the identification of vertices in a remarkable way by using various graph theory concepts including graph coloring, labeling of vertices, domination in graphs, vertex covering, graph automorphisms with symmetry breaking technique, independence of vertices, and by defining the metric on graphs, to name a few.
By defining the metric on a graph, the problem of identification of vertices attracted many researchers due to its significant applications in several extents including verification, security, and discovery of networks [1], the chemistry of pharmaceutics for drug designing [2], mastermind game strategies [3], navigation of robots [4], connected joins in graphs [5], and solution of coin weighing problems [6]. Because of these practical significances of this problem, from the last two decades, numerous researchers identified vertices of a graph by considering the metricrelated well-known concept of the metric dimension [2,7,8]. Later on, many researchers extended the study of this concept by defining its several variations including the fractional metric dimension [9], the resolving domination [10], the doubly metric dimension [11], the independent metric dimension [12], the weighted metric dimension [13], the k-metric dimension [14], the solid metric dimension [15], the mixed metric dimension [16], the local metric dimension [17], the simultaneous metric dimension [18], the strong metric dimension [5], and the connected metric dimension [19].
is paper is aimed to identify the vertices of three convex plane graphs by using geodesics between them, which provides the strong metric dimension of these graphs.

Geodesic Identification: The Strong Metric Dimension
Let G be a simple connected graph. A shortest path between two vertices x and y in G is known as a x − y geodesic. e geodesic identification of vertices is equivalently the strong resolvability of vertices, which is defined as follows: a vertex w of G performs the geodesic identification for the vertices in a pair (x, y) (i.e., w strongly resolves the vertices in a pair (x, y)) if either x belongs to (lies on) a y − w geodesic or y belongs to (lies on) a x − w geodesic. A set S of vertices in G is said to be a strong metric generator for G if for every pair of vertices of G, there is always a vertex in S which performs the geodesic identification for the vertices in the pair. e cardinality of such a smallest set S is called the strong metric dimension of G, denoted by sdim(G) [5]. Sebö and Tannier initiated the study of geodesic identification of vertices by defining the concepts of strong metric generator and strong metric dimension in 2004 [5]. Later on, this study was extended by many researchers, and they contributed to the literature with a variety of remarkable research work. To develop the readers interest, we shortly survey the strong metric dimension problem as follows: (i) e strong metric dimension problem has been solved for Sierpiński graph in [20], for hamming graphs in [21], for some convex polytopes in [22,23], for wheel related graphs (including n-fold wheel, sunflower, helm, and friendship graphs) in [24], for path, cycle, complete, complete bipartite, and tree graphs in [25], for Cayley graphs in [26], for Cartesian sum graphs in [27], for the power graph of a finite group in [28], for distance-hereditary graphs in [29], for generalized butterfly and starbarbell graphs in [30], for antiprism and king graphs in [31], for sun, windmill, and Möbius ladder graphs in [32], and for crossed prism in [33]. (ii) e strong metric dimension of various products of graphs including Cartesian product, direct product, strong product, lexicographic product, rooted product, and corona product has been supplied through the articles in [26,27,30,31,[33][34][35][36][37][38][39].
(iii) e fractional version of the strong metric dimension problem has been introduced in [40] and further studied for various graphs and graph products in [41,42].
(iv) e technique of the computation of strong metric dimension with the concept of the vertex cover number has been provided in [38] by proposing the construction of strong resolving graph of a connected graph. Furthermore, the article in [25] supplied some fundamental realizations and characterizations of the strong metric dimension problem in connection with the strong resolving graph.
(v) To solve the strong metric dimension problem, the genetic algorithmic approach is used in [43], and the variable neighborhood search method is used in [44]. (vi) e article in [45] supplied the Nordhaus-Gaddum type results whenever the strong metric dimension problem was solved for graphs and their compliments. (vii) To compute the strong metric dimension of graphs using optimization techniques, the integer linear programming model for the strong metric dimension problem was formulated in [22]. (viii) e complexity and the optimal approximability in the computation of the strong metric dimension problem of graphs have been discussed in [38,46]. (ix) Furthermore, we refer a survey in [47] and the Ph.D. thesis in [48] to the readers having interest in the computation of the strong metric dimension of graphs.
With this paper, we extend the study of the identification of vertices using the concept of strong metric dimension. We consider three convex plane graphs and investigate their strong metric dimension by performing the geodesic identification of vertices in the graphs.

Basic Works
Let G be a simple and connected graph with vertex set V(G) and edge set E(G). We denote the adjacent vertices u and v by u ∼ v and nonadjacent vertices by u≁v in G.
e number of vertices adjacent with a vertex v is called its degree and is denoted by where l is the length of (the number of edges in) a x − y geodesic. Accordingly, a vertex w of G performs the geodesic identification for the vertices in a pair (x, y) if and only if either Kratica et al. in [22] supplied the following two results, which are useful tools for the geodesic identification of vertices.
Lemma 1 (see [22]). Let (u, v) be a pair of distinct vertices in a connected graph G such that en, there is no vertex in V(G) − u, v { } which performs the geodesic identification for the pair (u, v).
Proposition 1 (see [22]). If S is a strong metric generator for a connected graphs G, then for every pair (u, v) of distinct vertices in G satisfying both the conditions in (1), either u ∈ S or v ∈ S.

Mathematical Problems in Engineering
For a connected graph G, the number is called the diameter of G, where ecc(v) is the eccentricity of a vertex v. e following result, provided by Kratica et al. in [22], is also a useful tool for the geodesic identification of vertices.
Proposition 2 (see [22]). If S is a strong metric generator for a connected graphs G, then for every pair (u, v) A vertex v of G for which d(v) � 1 is known as a leaf. e following result describes the geodesic identification of leaves in a connected graph.

Lemma 2.
If S is a strong metric generator for a connected graphs G and (u, v) is a pair of distinct leaves in G, then either u ∈ S or v ∈ S.
Proof. Since u and v are leaves, e identities (2)-(5) provide that the pair (u, v) satisfies both the conditions in (1), and hence, Proposition 1 yields the required result.
Lemma 2 supplies the following straightforward proposition. □ Proposition 3. If S is a strong metric generator for a connected graph G and L is the set of m leaves in G, then S must contain at least m − 1 leaves from L.

Convex Plane Graphs
A graph G is convex if a straight line joining any two vertices of G entirely lies within the region occupied by G. A graph G is a plane graph if it is free from crossing of edges.
roughout this section, it should be helpful to keep in mind the following points: (i) e right arrow ( ⟶ ) in the supper script of the notation of geodesics will indicate forward nature of geodesics (see Figure 1). (ii) e left arrow (←) in the supper script of the notation of geodesics will indicate backward nature of geodesics (see Figure 1).
(iii) e indices greater than n or less than 1 will be taken modulo n.
In the next sections, we investigate the strong metric dimension of three convex plane graphs.

Convex Plane Graph
e graph of a convex polytope S n is defined in [49] for n ≥ 3. e vertex set of S n is V(S n ) � a i , b i , c i , d i ; 1 ≤ i ≤ n , and its edge set is e convex plane graph S p n (p for pendant) can be obtained from the graph of a convex polytope S n by attaching one pendant vertex (leaf ) e i to the vertex d i of S n for each 1 ≤ i ≤ n (see Figure 2) [50]. us, the vertex and edge sets of We investigate the strong matrix dimension of S p n by supplying the following main result.
e next four lemmas will lead the proof of eorem 1. Proof. Let n � 2k + 1 with k ≥ 1, and consider Table 1 for In S p n , note the following points: P1: L � e i ; 1 ≤ i ≤ n is the set of leaves. P2: for each 1 ≤ i ≤ n, d(a i , e i+k ) � k + 4. It follows, according to Table 1, that the pair (a i , e i+k ) satisfies both the conditions in (1).
en, since the pair (a k+1 , e 1 ) satisfies P2 and P3, we have either a k+1 ∈ S or e 1 ∈ S, by Propositions 1 and 2.
Hence, Proof. Let n � 2k with k ≥ 2, and consider Table 2 for In S p n , the following points hold: Table 2 yields that the pairs (a i , e i+k− 1 ), (a i , e i+k ) and (b i , e i+k ) satisfy both the conditions in (1) for each 1 ≤ i ≤ n. Table 2 further implies that the pair (a j , a j+k ) satisfies both the con- Due to P1, without loss of generality, we let e 2 , e 3 , . . . , e n ∈ S, by Proposition 3.
en, as the pairs (a k+2 , e 1 ), (a k+1 , e 1 ) and (b k+1 , e 1 ) satisfy P2 and P3, we must have either a k+2 ∈ S or a k+1 ∈ S or b k+1 ∈ S or e 1 ∈ S, by Propositions 1 and 2. For the geodesic identification of these three pairs, it is enough to consider e 1 in S. Furthermore, for each 1 ≤ j ≤ k, either a j ∈ S or a j+k ∈ S, due to P2 and Proposition 1.
Proof. For any s ∈ S and any u ∈ V(S p n ) − s { }, since the vertex s performs the geodesic identification for the pair (s, u), we have to perform the geodesic identification for every pair (x, y) of distinct vertices with x, y ∈ V(S p n ) − S. For each 1 ≤ i ≤ n, consider the following a i − e i+k geodesics of length k + 4: Vertex

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Each geodesic, listed from (9) to (16), provides that the vertex e i+k ∈ S performs the geodesic identification for every Proof. We have to perform the geodesic identification each possesses the geodesic identification by the vertex s. For each 1 ≤ i ≤ n, consider the following a i − e i+k− 1 geodesics and a i − e i+k geodesics of length k + 3: Except the pairs (a i , a j ) for k ≤ i ≤ n − 1 and i + 1 ≤ j ≤ n, all the pairs of vertices owned the geodesic identification by the vertices e i+k and e i+k− 1 due to the geodesics listed from (17) to (24). By considering the following a k − a n geodesic P → : a k ∼ a k+1 ∼ a k+2 ∼ · · · ∼ a n− 1 ∼ a n , we get that the vertex a k performs the geodesic identification for all the remaining pairs (a i , a j ). Hence, S is a strong metric generator for S p n .
□ Proof ( eorem 1). e lower bound for the strong metric dimension of S p n is provided by Lemmas 3 and 4, whereas the upper bound is due to Lemmas 5 and 6. e graph of a convex polytope T n is defined in [49] for n ≥ 3. e vertex set of T n is V(T n ) � a i , b i , c i , d i ; 1 ≤ i ≤ n , and its edge set is Vertex Mathematical Problems in Engineering 5 e convex plane graph T p n (p for pendant) can be obtained from the graph of a convex polytope T n by attaching one pendant vertex (leaf ) e i to the vertex d i of T n for each 1 ≤ i ≤ n (see Figure 3) [50]. Accordingly, the vertex and edge sets of Mainly, we have the following result for the strong metric dimension of T p n .
Theorem 2. For n ≥ 3, let T p n be a convex plane graph. en, sdim T p n � 2n, when n is odd, 5n 2 , when n is even.
e proof of eorem 2 will be followed after the next four lemmas.

Lemma 7.
For odd values of n ≥ 3, if S is a strong metric generator for T p n , then |S| ≥ 2n.
Proof. Let n � 2k + 1 with k ≥ 1, and consider Table 3 for Note the following three points in T p n : P1: L � e i ; 1 ≤ i ≤ n is the set of leaves. P2: d(a i , e i+k− 1 ) � d(a i , e i+k ) � d(b i , e i+k ) � k + 3 and d(a i , c i+k ) � k + 2 for each 1 ≤ i ≤ n. Hence, Table 3 ensures that pairs of vertices in T p n satisfying both the conditions in (1) are (a i , e i+k− 1 ), (a i , e i+k ), (b i , e i+k ), and (a i , c i+k ) for each 1 ≤ i ≤ n.
By Proposition 3, let us take e 2 , e 3 , . . . , e n ∈ S without loss of generality due to P1. en, from the remaining pairs (a k+3 , e 1 ), (a k+2 , e 1 ), and (b k+2 , e 1 ), we must have either a k+3 ∈ S or a k+2 ∈ S or b k+2 ∈ S or e 1 ∈ S, by Propositions 1 and 2, because these pairs are satisfying P2 and P3. Here, it can be seen, by letting e 1 ∈ S only, that all these pairs possess the geodesic identification by the vertex e 1 . Furthermore, for each 1 ≤ i ≤ n, from the pair (a i , c i+k ) satisfying P2, either a i ∈ S or c i+k ∈ S, by Proposition 1. Hence, |S| ≥ n − 1 + 1 + n � 2n. Proof. Let n � 2k with k ≥ 2, and consider Table 4 for e following points are considerable in T p n : P1: L � e i ; 1 ≤ i ≤ n is the set of leaves. P2: d(a i , e i− 1 ) � 4, d(a i , e i+k− 1 ) � k + 3 and d(b i , c i+k ) � k + 1, which implies according to Table 4, that the pairs (a i , e i− 1 ), (a i , e i+k− 1 ) and (b i , c i+k ) satisfy both the conditions in (1) for each 1 ≤ i ≤ n. Moreover, for each 1 ≤ j ≤ k, d(a j , a j+k ) � k, and so Table 4 concludes that the pair (a j , a j+k ) also satisfies both the conditions in (1).
Now, without loss of generality, let us take e 2 , e 3 , . . . , e n in S, by Proposition 3 and due to P1. en, from the remaining pairs (a k+2 , e 1 ) and (a 2 , e 1 ) satisfying P2 and P3, respectively, we must have either a k+2 ∈ S or e 1 ∈ S by Proposition 2, and either a 2 ∈ S or e 1 ∈ S by Proposition 1. If we take e 1 in S, then this vertex performs the geodesic identification for both the remaining pairs. Also, Proposition 1 yields that either b i ∈ S or c i+k ∈ S for each 1 ≤ i ≤ n and either a j ∈ S or a j+k ∈ S for each 1 ≤ j ≤ k because of the point P2. erefore, |S| ≥ n − 1 + 1 + n + k � (5n/2). □ Lemma 9. For n � 2k + 1 with k ≥ 1, the set S � a 1 , a 2 , . . . , a n , e 1 , e 2 , . . . , e n ⊂ V(T All other pairs of vertices possess the geodesic identification by the vertices e i+k− 1 and e i+k using a i − e i+k− 1 geodesics and a i − e i+k geodesics of length k + 3, listed from the following equations: Mathematical Problems in Engineering Now, for each 1 ≤ i ≤ n, by considering b i − e i+k geodesics, listed from (38) to (43), the vertex e i+k performs the geodesic identification for all the pairs of vertices listed in (29): Vertex Vertex It concludes that S is a strong metric generator for T p n . □ Lemma 10. If n � 2k with k ≥ 2, then the set S � a 1 , a 2 , . . . , a k , b 1 , b 2 , . . . , b n , e 1 , e 2 , . . . , e n ⊂ V T p n , is a strong metric generator for T p n .
Proof. Since the vertex s ∈ S performs the geodesic identification for every pair of vertices including s, it implies that we have to perform the geodesic identification by the vertices in S for all the pairs of vertices from V(T p n ) − S. For this, consider the following a i − e i+k− 1 geodesics of length k + 3: Due to geodesics, listed from (45) to (50), the vertex e i+k− 1 performs the geodesic identification for all the pairs of vertices except the pairs (c i , c i+k ), (d i , c i+k ), (d i , d i+k ) for all 1 ≤ i ≤ n and the pair (a j , d j− 1 ) for each k + 1 ≤ j ≤ n. ese pairs are identified as follows: (i) For each 1 ≤ i ≤ n, the vertex b i performs the geodesic identification for the pair (c i , c i+k ) due to the geodesic Q i �→ (iii) For each 1 ≤ i ≤ n, the vertex e i+k performs the geodesic identification for the pair (d i , d i+k ) due to the geodesic S i → : (iv) For each k + 1 ≤ j ≤ n, the vertex e j− 1 performs the geodesic identification for the pair (a j , d j− 1 ) due to the geodesic T i Hence, we conclude that S is a strong metric generator for T p n .  e graph of a convex polytope U n is defined in [49] for n ≥ 3. e vertex set of U n is V(U n ) � a i , b i , c i , d i , e i ; 1 ≤ i ≤ n , and its edge set is (51) e convex plane graph U p n (p for pendant) is obtained from the graph of a convex polytope U n by attaching one pendant vertex (leaf ) f i to the vertex e i of U n for each 1 ≤ i ≤ n (see Figure 4) [50]. us, we have the followings vertex and edge sets for U p n : e following main result provides the strong metric dimension of U p n . 8 Mathematical Problems in Engineering Theorem 3. For n ≥ 3, let U p n be a convex plane graph. en, 2n, when n is odd, 5n 2 , when n is even.
e proof of eorem 3 will be followed after the next four lemmas.

Lemma 11.
For odd values of n ≥ 3, if S is a strong metric generator for U p n , then |S| ≥ 2n.
Proof. Let n � 2k + 1 with k ≥ 1, and consider Table 5 for Note the following points in U p n : P1: L � f i ; 1 ≤ i ≤ n is the set of leaves. P2: for each 1 ≤ i ≤ n, the pairs of vertices (a i , f i+k ) and (c i , d i+k ) satisfy both the conditions in (1) because of Table 5 and since d( Due to P1 and by Proposition 3, we let f 2 , f 3 , . . . , f n ∈ S without loss of generality. en, the remaining pair (a k+2 , f 1 ) satisfies both P2 and P3. So, we must have either a k+2 ∈ S or f 1 ∈ S, by Propositions 1 and 2. Furthermore, Proposition 1 yields that, from the pair (c i , d i+k ), either c i ∈ S or d i+k ∈ S for all 1 ≤ i ≤ n due to P2. Hence, |S| ≥ n − 1 + 1 + n � 2n. Proof. Let n � 2k with k ≥ 2, and consider Table 6 for 1 ≤ i ≤ n. e following points hold in U p n : It follows, by Table 6, that the pairs of vertices (a i , . erefore, Table 6 again yields that the pairs of vertices (c j , c j+k ) and (d j , d j+k ) satisfy both the conditions in (1) for all Without loss of generality, taking f 2 , f 3 , . . . , f n in S, by Proposition 3 and due to P1, we get three pairs of vertices (a k+2 , f 1 ), (a k+1 , f 1 ), and (d k+1 , f 1 ), which are still satisfying P2 and P3. For these pairs, Propositions 1 and 2 allow us to take either a k+2 ∈ S or a k+1 ∈ S or f 1 ∈ S. It can be seen that the best suitable choice, to perform the geodesic identification for these three pairs, is to take f 1 in S. For otherwise, we will get |S| ≥ 3n. Furthermore, the point P3 together with Proposition 2 yields that, from the pairs of vertices (a i , c i+k ) and (c i , d i+k ), either a i ∈ S or c i+k ∈ S and c i ∈ S or d i+k ∈ S for all 1 ≤ i ≤ n. Accordingly, we discuss the following four cases: (i) If a i ∈ S and c i ∈ S for all 1 ≤ i ≤ n, then we must have either d j ∈ S or d j+k ∈ S for all 1 ≤ j ≤ k, by the point P2 and Proposition 1. For otherwise, we get the pair of vertices (d j , d j+k ), for all 1 ≤ j ≤ k, which is left unidentified by the elements of S, but we have |S| ≥ n − 1 + 1 + n + n + k � (7n/2). (ii) If a i ∈ S and d i+k ∈ S for all 1 ≤ i ≤ n, then we must have either c j ∈ S or c j+k ∈ S for all 1 ≤ j ≤ k, by the point P2 and Proposition 1. Otherwise, we get the pair of vertices (c j , c j+k ), for all 1 ≤ j ≤ k, which is left unidentified by the elements of S, but we have |S| ≥ n − 1 + 1 + n + n + k � (7n/2). (iii) If c i+k ∈ S, then of course c i ∈ S for all 1 ≤ i ≤ n.
Moreover, we must have either d j ∈ S or d j+k ∈ S for  Vertex Vertex all 1 ≤ j ≤ k, by the point P2 and Proposition 1. For otherwise, we get the pair of vertices (d j , d j+k ), for all 1 ≤ j ≤ k, which is left unidentified by the elements of S. en, we have |S| ≥ n − 1 + 1 + n + k � (5n/2). (iv) If c i+k ∈ S and d i+k ∈ S for all 1 ≤ i ≤ n, then we have |S| ≥ n − 1 + 1 + n + n � 3n.
It can be concluded from these four cases that the most suitable choice to construct a strong metric generator S with minimum cardinality is found in the Case 3, which yields that |S| ≥ (5n/2). □ Lemma 13. If n � 2k + 1 with k ≥ 1, then the set S � c 1 , c 2 , . . . , c n , f 1 , f 2 , . . . , f n ⊂ V(U p n ) is a strong metric generator for U p n .
Proof. It is enough to perform the geodesic identification for those pairs of vertices of U p n having no element from the set S because every pair of vertices having one element s from S possesses the geodesic identification by the element s. Let us consider the following a i − f i+k geodesics of length k + 5, for each 1 ≤ i ≤ n: e vertex f i+k performs the geodesic identification for all the pairs of vertices due to geodesics listed from (54) to (59), except the following pairs of vertices: For 1 ≤ i ≤ n, 2 ≤ j ≤ k, the following geodesics ensure the geodesic identification of the pairs, given in (60) and (61), by some vertices of S as described in Table 7: Hence, we conclude that S is a strong metric generator for U p n . erefore, we should perform the geodesic identification for each pair (x, y) of vertices of U p n by the vertices in S for both x, y∈S. For each 1 ≤ i ≤ n, the following a i − f i+k− 1 and a i − f i+k geodesics of length k + 4 are useful for our purpose: It can be seen that, for some pairs of vertices, the vertex f i+k− 1 performs the geodesic identification due to geodesics in (67)-(69), and for some pairs of vertices, the vertex f i+k performs the geodesic identification due to geodesics in (70)-(72). e pairs of vertices, which are left unidentified, are as follows: e j , d j+k , e j , e j+k , for 1 ≤ j ≤ k, d r , d r+1 , for k + 1 ≤ r ≤ n − 1, d l , d l+m , for k + 1 ≤ l ≤ n − 2, 2 ≤ m ≤ k − 1.
□ Proof ( eorem 3). e lower bound for the strong metric dimension of U p n is provided by Lemmas 11 and 12, whereas the upper bound is due to Lemmas 13 and 14. Table 7: Geodesic identification for the pairs given in (60) and (61).

Pairs
Vertex performing the geodesic identification Due to the geodesic in

Pairs
Vertex performing the geodesic identification Due to the geodesic in (a i , a i+k ), (a i , b i+k )