Local fractional metric dimension of rotationally symmetric planar graphs arisen from planar chorded cycles

Rend. Mat. Appl. (7), Volume 44 (2023), 159–179 RENDICONTI DI MATEMATICA E DELLE SUE APPLICAZIONI Local fractional metric dimension of rotationally symmetric planar graphs arisen from planar chorded cycles Shahbaz Ali, Ra´ul M. Falc´on and Muhammad Khalid Mahmood Abstract. In this paper, a new family of rotationally symmetric planar graphs is described based on an edge coalescence of planar chorded cycles. Their local fractional metric dimension is established for those ones arisen from chorded cycles of order up to six. Their asymptotic behaviour enables us to ensure the existence of new families of rotationally symmetric planar graphs with either constant or bounded local fractional dimension. 1. Introduction In the 1970’s, Slater [35] and Harary and Melter [19] introduced independently the metric dimension problem, which consists of determining the minimum number of vertices within a graph that may uniquely be represented by their respective vector of distances. Being NP-hard [17], this problem has explicitly been solved for different types of graphs [8,20,23]. General and specific bounds are known depending on the order, maximum degree or diameter of the graph under consideration [8,14,36]. In this regard, Imran et al. [21,22] asked for characterizing families of (rotationally symmetric) planar graphs with constant metric dimension (see also [8,24,26]). The metric dimension problem plays a relevant role not only in the study of structural properties of graphs, but also in solving real life problems such as robot navigation [26], pattern recognition and image processing [30], representation of chemical compounds [8], combinatorial optimization [33] or networking [5], amongst others. Particularly, the metric dimension problem concerning hexagonal graphs [29,34,38] has acquired special relevance because of their implementation in computer graphics [27], multiprocessor networks [9] and cellular networks [16]. In 2000, Chartrand et al. [8] formulated the metric dimension problem as an integer programming problem. Shortly after, Currie and Oellermann [10] formulated a linear programming relaxation whose optimal solution they termed fractional metric dimension of the graph (see also [3,11]). This new problem has explicitly been solved for different types of graphs [4,12,13,15,32]. In 2018, a local version of the fractional metric dimension concerning only adjacent vertices was introduced by Benish et al. [7] (see also [6]). Even if its study 2020 Mathematics Subject Classification: 05C72, 05C12, 05C10. Keywords: Local fractional metric dimension, rotationally symmetric planar graph, planar chorded cycle, local resolving neighbourhood. 160 A. Shahbaz, R. M. Falc´on and M. K. Mahmood is still in a very initial stage, the local fractional metric dimension has explicitly been determined for several types of graphs [1,25]. More recently, Liu et al. [28] have computed the local fractional metric dimension of a family of rotationally symmetric planar graphs derived from an edge coalescence of a cycle of order m with mdistinct chorded cycles of a same order n∈ {3,4,5}. This paper delves into this last topic by describing a new family of rotationally symmetric planar graphs arisen from an edge coalescence of mplanar chorded cycles of order n. The paper is organized as follows. In Section 2, we remind some preliminary concepts and results on graph theory that are used throughout the paper. In Section 3, we describe the mentioned family of rotationally symmetric planar graphs. We focus in particular on those ones derived from planar chorded cycles of order n∈ {4,5,6}, for which we study their local fractional metric dimension. The case n= 4 is dealt with at the end of Section 3, whereas exhaustive analyses of the cases n= 5 and n= 6 are done in Section 4. The obtained results are summarized in the conclusion section, where we make use of the asymptotic behaviour in order to ensure which ones of these rotationally symmetric planar graphs have either a constant or a bounded local fractional metric dimension. Finally, all the tables described in the body of the manuscript are enumerated in Appendix A. 2. Preliminaries Let us review some basic concepts and results on graph theory that are used throughout the paper. See [37] for more details about this topic. Any graph Gis formed by a set V(G) of vertices and a set E(G) of edges so that each edge contains two vertices, which are said to be adjacent. The subset of vertices that are adjacent to a given vertex v∈V(G) constitutes its neighborhood N(v). Two adjacent vertices having the same neighborhood are called true twin vertices. From here on, we denote uv the edge formed by two vertices u, v ∈ V(G). The number of vertices and the number of edges of the graph Gconstitute, respectively, its order and size. If both of them are finite, then the graph is said to be finite. The graph Gis called bipartite if the set V(G) may be partitioned into two subsets so that every edge contains exactly one vertex of each subset. Further, two graphs are isomorphic if there exists a bijection between their sets of vertices preserving their adjacency. Apath between two distinct vertices v, w ∈V(G) is any ordered sequence of adjacent and pairwise distinct vertices hv0=v, v1, . . . , vn−2, vn−1=wiin V(G), with n > 2. A graph is connected if there always exists a path between any pair of vertices. If the initial and final vertices of a path coincide, then it is called a cycle. If all the vertices of a cycle are joined to a new vertex, then the resulting graph is called a wheel. The new vertex is called the center of the wheel. Further, a chord of an existing cycle in Gis any edge of the graph containing two non-adjacent vertices of the cycle. A chorded cycle is any cycle containing at least one chord. Aplanar graph is any graph that can be embedded in the plane. That is, it may be drawn on the plane without crossing edges. Figure 1illustrates the set of Local fractional metric dimension of symmetric planar graphs 161 non-isomorphic planar chorded cycles of order n≤6. Figure 1: Non-isomorphic planar chorded cycles of order n≤6. The union of two graphs G1and G2is the graph G1∪G2such that V(G1∪G2) = V(G1)∪V(G2) and E(G1∪G2) = E(G1)∪E(G2). If E(G1)6=∅ 6=E(G2), then the edge coalescence [18] of G1and G2via u1v1∈E(G1) and u2v2∈E(G2) is the graph G1·G2(u1v1, u2v2:u3v3) resulting after identifying in G1∪G2both edges u1v1and u2v2, which merge into a single new edge u3v3. Figure 2illustrates this concept with three different examples of edge coalescence between the same pair of planar chorded cycles of order six. Figure 2: Edge coalescences between the same pair of planar chorded cycles of order six. Let G= (V(G), E(G)) be a finite connected graph. The minimum number of edges within a path between two vertices v, w ∈V(G) constitutes the distance d(u, v). The diameter diam(G) of the graph Gis the maximum distance between any two vertices in V(G). Further, the representation of a vertex v∈V(G) with respect to an ordered subset S={v1, . . . , vk} ⊆ V(G) is the ordered k-tuple r(v|S) := (d(v, v1), . . . , d(v, vk)). The subset Sis a resolving set for the graph Gif r(v|S)6=r(w|S), for every pair of distinct vertices v, w ∈V(G). If such condition holds for every edge vw ∈E(G), then the subset Sis a local resolving set. The (local) metric dimension of the graph Gis then defined as the minimum number of vertices contained in any of its (local) resolving sets. It is denoted dim(G) (ldim(G) in its local version). The resolving neighbourhood of a pair of vertices v, w ∈V(G) is the set R{v, w}:= {u∈V(G)|d(v, u)6=d(w, u)}. From here on, in order to simplify the notation of this manuscript, we also denote RC{v, w}=V(G)\ R{v, w}. A resolving function [3] of the graph Gis any map ϑ:V(G)→[0,1] such that X u∈R{v,w} ϑ(u)≥1,(2.1) 162 A. Shahbaz, R. M. Falc´on and M. K. Mahmood for every pair of distinct vertices v, w ∈V(G). The fractional metric dimension of the graph Gis dimf(G) := min   X v∈V(G) ϑ(v): ϑis a resolving function of G   . The concepts of local resolving neighbourhood and local resolving function arise similarly in case of dealing only with pairs of adjacent vertices. Then, the local fractional metric dimension of the graph Gis defined as ldimf(G) := min   X v∈V(G) ϑ(v): ϑis a local resolving function of G   . Further, we denote from now on `(G) := min{|R {v, w}| :vw ∈E(G)}. In particular, since v, w ∈ R{v, w}, for all v, w ∈V(G), it is `(G)≥2. The next result follows from all the previous definitions. Lemma 2.1 ([6,7,31]).Let Gbe a finite connected graph of order n≥2. Then, ldimf(G)≤dimf(G),(2.2) n n−ldim(G)+1 ≤ldimf(G)≤n `(G)≤n 2(2.3) 1≤ldimf(G)≤ldim(G)≤n−diam(G).(2.4) In addition, the following assertions are satisfied. 1. ldimf(G)=1if and only if the graph Gis bipartite. 2. ldimf(G) = n 2if and only if each vertex in V(G)has a true twin vertex. Example 2.2. Let Gbe the planar graph described in Figure 3. In order to determine an upper bound of ldimf(G), we determine the cardinality of its local resolving neighbourhoods. To this end, the symmetry of the graph Genables us to focus on the following four resolving neighbourhoods. RC{v1, v7}={v10, v14, v15, v16, v20, v21}. RC{v7, v13}={v2, v5, v8, v12}. RC{v13, v18}={v3, v6, v21, v24}. RC{v13, v24}={v4, v5, v10, v11, v12, v16, v17, v18, v22, v23}. Thus, `(G) = 14 and hence, Condition (2.3) implies that ldimf(G)≤24 14 =12 7. In fact, a simple computation establishes that ldimf(G) = 3 2. Local fractional metric dimension of symmetric planar graphs 163 Figure 3: Planar graph Gsuch that ldimf(G) = 3 2≤12 7. 3. A family of rotationally symmetric planar graphs arisen from planar chorded cycles This paper focuses on the asymptotic behaviour of the local fractional metric dimension of a particular family of rotationally symmetric planar graphs. They arise from a sequential edge coalescence among a series of disjoint copies of a given planar chorded cycle. In this section, we detail their construction and establish some basic results concerning their local fractional metric dimension. Let G1, . . . , Gmbe mdisjoint copies, with m≥2, of a planar chorded cycle G of order n, whose set of vertices is V(G) = {v1, . . . , vn}. Let vk i∈V(Gk) denote the corresponding copy of each vertex vi∈V(G). Without loss of generality, we assume that the vertices are naturally labeled counterclockwise. Then, we are interested in the planar graph Gm(G) that is sequentially defined as follows. •Let G1(G) := G1·G2(v1 2v1 3, v2 n−1v2 n:v2 n−1v2 n). (Notice that we label the merged new edge in the same way that the edge in the second graph under consideration. The same is done in the subsequent steps.) •Let Gk(G) := Gk−1(G)·Gk+1(vk 2vk 3, vk+1 n−1vk+1 n:vk+1 n−1vk+1 n), for each positive integer k∈ {2, . . . , m −1}. •Finally, let Gm(G) := Gm−1(G)· Gm−1(G)(vm 2vm 3, v1 n−1v1 n:v1 n−1v1 n). The resulting graph Gm(G) is a rotationally symmetric planar graph of order m·(n−2). Based on the planar chorded cycles described in Figure 1, we enumerate in Figure 4all the vertex-labeled planar chorded cycles of order n≤6 on which the just defined constructive procedure may be implemented in order to get nonisomorphic rotationally symmetric planar graphs. From here on, we refer them as 164 A. Shahbaz, R. M. Falc´on and M. K. Mahmood quadrilateral (Q1and Q2, for n= 4), pentagonal (P1to P6, for n= 5) or hexagonal (H1to H17, for n= 6) chorded cycles. Figure 4: Quadrilateral, pentagonal and hexagonal chorded cycles. Figure 5illustrates a representation of each one of the two non-isomorphic rotationally symmetric planar graphs arisen from the quadrilateral chorded cycles Q1and Q2. In addition, Figure 6illustrates the representations of a pair of rotationally symmetric planar graphs arisen from the pentagonal chorded cycle P4 and the hexagonal chorded cycle H5. Notice that the first and last vertical edges in each one of these two representations refer to the same edge. In a similar way, Figures 7and 8outline all the rotationally symmetric planar graphs arisen from pentagonal and hexagonal chorded cycles. Notice that, even if the vertex-labeling has been omitted for making clearer the illustration, it coincides in each case with that one described in Figure 6. Table 1illustrates the local fractional metric dimension of each one of the planar chorded cycles described in Figure 4. Notice in particular that the minimum value is reached for both graphs H1and H2, which constitute different vertexlabelings of the same bipartite graph (see Lemma 2.1). In order to determine the Local fractional metric dimension of symmetric planar graphs 165 Figure 5: The graphs Gm(Q1) and Gm(Q2). Figure 6: The graphs Gm(P4) and Gm(H5) remaining values, we have computationally solved the linear programming problem associated to each case [10]. Example 3.1. The linear programming problem based on the local fractional metric dimension of the hexagonal chorded cycle H5consists of finding a map ϑ:V(H5)→[0,1] that minimizes P1≤i≤6ϑ(vi) subject to          P1≤i≤6ϑ(vi)≥1, ϑ(v1) + ϑ(v2) + ϑ(v3) + ϑ(v4) + ϑ(v6)≥1, ϑ(v3) + ϑ(v4) + ϑ(v5) + ϑ(v6)≥1, ϑ(v1) + ϑ(v2) + ϑ(v5)≥1. Since R{v1, v2}=R{v2, v3}=R{v3, v4}=V(H5), these three edges are related to the case P1≤i≤6ϑ(vi)≥1. The second case follows from R{v1, v4}and R{v1, v6}. The third one follows from R{v4, v5}and R{v5, v6}). Finally, the fourth one follows from R{v1, v5}. An optimal solution of this problem is the resolving function 166 A. Shahbaz, R. M. Falc´on and M. K. Mahmood Figure 7: Rotationally symmetric planar graphs arisen from pentagonal chorded cycles. ϑof H5satisfying that ϑ(vi) = 1 2, if i∈ {1,3,5}, and zero, otherwise. Hence, ldimf(H5) = 3 2. We are interested in the asymptotic behaviour of the local fractional metric dimension of each rotationally symmetric planar graph arisen from the planar chorded cycles described in Figure 4. In this regard, we finish this section by dealing with those graphs arisen from quadrilateral chorded cycles. Those ones arisen from pentagonal and hexagonal chorded cycles are studied in Section 4. Firstly, we prove a preliminary lemma concerning the local fractional metric dimension of a wheel. (Notice that it differs from the unproven Theorem 3 in [2], whose third assertion seems not to be true.) Lemma 3.2. Let Wnbe the wheel graph of order n≥4. Then, ldimf(Wn) =      2,if n= 4, 3 2,if n∈ {5,6}, n−1 4,otherwise. Proof. The case n≤6 follows directly from solving the corresponding linear programming problem. In any case, notice that the case n= 4 holds readily from Lemma 2.1 once it is observed that every pair of vertices within the wheel graph W4are true twin. Now, in order to deal with the case n > 6, let us suppose that V(Wn) = {v0, . . . , vn−2, w}. Here, wdenotes the center of the wheel graph Wn. In addition, vivi+1 ∈E(Wn), for every non-negative integer i<n, where, from here on, all the indices are taken modulo (n−1). Then, the computation of ldimf(Wn) requires to minimize ϑ(w) + P0≤i≤n−2ϑ(vi) subject to (ϑ(vi−1) + ϑ(vi) + ϑ(vi+1) + ϑ(vi+2)≥1, ϑ(w)−ϑ(vi−1)−ϑ(vi+1) + P0≤i≤n−2ϑ(vi),∀i≤n−2. Local fractional metric dimension of symmetric planar graphs 167 Figure 8: Rotationally symmetric planar graphs arisen from hexagonal chorded cycles. The first case derives from R{vi, vi+1}, whereas the second one derives from R{w, vi}. An optimal solution of this problem is the resolving function ϑof the wheel graph Wnsatisfying that ϑ(w) = 0 and ϑ(vi) = 1 4, if 0 ≤i≤n−2. Hence, ldimf(Wn) = n−1 4. The following result establishes the local fractional metric dimension of Gm(Q1) and Gm(Q2). Proposition 3.3. Let m≥2be a positive integer. Then, ldimf(Gm(Q1)) = (3 2,if m= 2, m 2,otherwise.ldimf(Gm(Q2)) = (3 2,if m≤4, m 4,otherwise. Proof. For Q1, the result holds from Lemma 3.2, because the planar graph Gm(Q1) is a wheel graph of order 2m+ 1. For Q2, the case m∈ {2,3,4}follows directly from solving the corresponding linear programming problem. In order to deal with m≥5, the computation of ldimf(Gm(Q2)) requires to solve the linear programming problem consisting on minimizing the objective function ϑ(v1 3) + P1≤i≤m(ϑ(vi 1) + ϑ(vi 4)) so that the following cases hold, for all i≤m. Here, the superscripts are all of them taken modulo m. •From R{vi 1, vi 4}, it is ϑ(v1 3) + Pj6∈{i+1, i+2}ϑ(vj 1) + ϑ(vj 4)≥1. •From R{v1 3, vi 4}, it is ϑ(v1 3) + Pj6∈{i−2, i+1}ϑ(vj 1) + Pj6∈{i−1, i+1}ϑ(vj 4)≥1. •From R{vi 4, vi+1 4}, it is ϑ(vi−2 1) + ϑ(vi−1 1) + ϑ(vi+1 1) + ϑ(vi+2 1) + ϑ(vi−1 4) + ϑ(vi 4) + ϑ(vi+1 4) + ϑ(vi+2 4)≥1. 174 A. Shahbaz, R. M. Falc´on and M. K. Mahmood ©The copyright of this article is retained by the Author(s). Open Access. This article is published in open access form and licensed under the terms of the Creative Commons Attribution–NonCommercial–ShareAlike 4.0 International (CC BY-NC-SA 4.0) (http://creativecommons.org/licenses/by-nc-sa/4.0/). Appendix A. Tables G ldimf(G) Qi(1 ≤i≤2) 3/2 Pi(1 ≤i≤6) 3/2 Hi(1 ≤i≤2) 1 Hi(3 ≤i≤12) 3/2 Hi(13 ≤i≤17) 5/3 Table 1: Local fractional dimension of planar chorded cycles of order n≤6. n abcd RC{vb a, vd c}Case 1 1142 {12,1m, 52,5m} 4151 1151 {42}m= 3 {1m1,4m3,5m3}m > 3 odd ∅m > 3 even 1152 {4i: 1 ≤i≤m}m∈ {2,3} {1i, 5i:i∈ {1,2,3, m} }Cm > 3 4142 {1m1,4m3,5m3}modd ∅meven 2 1151 {11,1i, 41,4i, 51,5i:m1< i ≤m}C 4142 {11,4m3,5m3} 5152 4151 ∅ Table 2: Resolving neighbourhoods of Gm(Pn), for n∈ {1,2}. Local fractional metric dimension of symmetric planar graphs 175 n abcd RC{vb a, vd c}Case 3 1151 {42}m∈ {2,3} {1i, 42,4i, 5i: 3 ≤i≤m1}m > 3 odd {1i, 42,4j, 5j: 3 ≤i≤m0,3≤j≤m3}m > 3 even 1152 ∅m= 2 {43}m= 3 {1i, 4m3,4(i+ 1),5(i+ 1): m3≤i<m}m > 3 odd {5m4}m > 3 even 4142 {11,12,52}m= 2 {11,12,43,52}m= 3 {11,12,1m3,4m3,52}m > 3 odd {11,12,4m4,52}m > 3 even 4151 ∅m= 2 {42,52}m= 3 {1i, 4m, 4i, 5i, 5m:m3≤i<m}m > 3 odd {1i, 4m0,4(i+ 1),5(i+ 1): m0< i < m}m > 3 even 4152 {42,53}m= 3 {1i, 42,4i, 53,5(i+ 1): 3 ≤i≤m1}m > 3 odd {1i, 42,4i, 5i: 3 ≤i≤m3}m > 3 even 4 1141 {12,1i, 51,5i, 42,4i: 3 ≤i≤m1} 1151 {1i, 41,4i, 4m, 5i:m3≤i<m} 4142 {4m3,5m3} 4151 {11,1m, 52,5m} 5 1141 {12,1m, 51,52} 1151 {41}m= 2 {1i, 41,42,4i, 5i: 3 ≤i≤m1}modd {1i, 41,42,4i, 5m3,5i: 3 ≤i≤m0}m > 2 even 1152 {1m3,1i, 41,4m3,4i, 4m, 5i, 5m:m3< i < m}modd {1i, 41,4i, 4m, 5i, 5m:m4≤i<m}meven 4142 {51}m= 2 {1m3,4m3,51}modd {51,5m4}m > 2 even 4151 {11,1i, 4i, 4m, 5i, 5m:m3≤i<m}Cmodd {11,1i, 4m3,4j, 5j:m0< i < m, m3< j ≤m}Cmeven 4152 {11,1i, 42,4i, 5i, 5m3: 3 ≤i≤m1}modd {11,1i, 42,4i, 5i: 3 ≤i≤m3}meven 6 1151 {1i, 4i, 5i: 2 ≤i≤m1} 1152 {1m3,1i, 4i, 51,5i:m3< i ≤m} 4142 {11,12,43,52}m= 3 {11,12,1m3,4m3,52}m6= 3 4151 {1i, 4j, 5i: 2 ≤i≤m1, m3≤j≤m} 4152 {1m3,1i, 4i, 51,5j:m3< i ≤m, 2≤j≤m1} 5152 {11,41,5m3} Table 3: Resolving neighbourhoods of Gm(Pn), for 3 ≤n≤6. 176 A. Shahbaz, R. M. Falc´on and M. K. Mahmood n abcd RC{vb a, vd c}Case 3 1151 {1i, 4j, 5j, 61,6i: 2 ≤j≤m1< i ≤m}modd {1i, 4j, 5k, 61,6l: 2 ≤j≤m0< i ≤m, 2≤k≤m3< l ≤m}meven 1161 {41,4m, 51} 1162 {41,5m1}modd {41,4(m0+ 1)}meven 4151 {62}modd {1m3,62}meven 4152 {62,6m3}modd {1m3,62}meven 5161 {11,1i, 4m1,4j, 5j, 5m, 6i: 2 ≤i≤m1< j < m}modd {1i, 4j, 5j, 5m, 6k: 2 < i ≤m0< j < m, 2≤k≤m3}meven 4 1161 {4i, 5i: 2 ≤i≤m1}modd {4i, 5j: 2 ≤i≤m0,2≤j≤m3}meven 4151 {ij :i∈ {1,4,5,6},2≤j≤m1}modd {1i, 4i, 5j, 6j: 2 ≤i≤m0,2≤j≤m3}meven 5152 {11,41,5m1,6m1}modd {11,1(m0+ 1),41,4m3}meven 5161 {62,6m}m= 2 {12,1(m−1),62,6m}m > 2 5 1141 {1i, 4j, 51,5j, 6k: 2 ≤i≤m1< j ≤m, 2< k ≤m3}modd {12,1i, 4j, 51,5k, 6i: 2 < i ≤m3< k ≤m, m0< j ≤m}meven 1151 {1i, 41,4j, 5j, 61,6i: 2 ≤j≤m1< i ≤m}modd {1m3,1i, 4j, 5k, 61,6i: 2 ≤k≤m3< i ≤m, 1≤j≤m0}meven 1161 {4m, 51}modd {4m, 51,5m3}meven 1162 {5m3}modd {4m3}meven 4151 {11,62,6m3}modd {11,1m3,62}meven 4152 {1m3}modd {6m4}meven 5161 {11,1i, 4m1,4j, 5j, 5m, 6i, : 2 ≤i≤m1< j < m}modd {1i, 4j, 5k, 6l: 1 ≤i≤m0< k ≤m, 2≤l≤m3≤j < m}meven 6 4161 {11,1m, 41,61}C Table 4: Resolving neighbourhoods of Gm(Hn), for 3 ≤n≤6. Local fractional metric dimension of symmetric planar graphs 177 n abcd RC{vb a, vd c}Case 7 1161 {12,1i, 4i, 4(m1+ 1),5i, 62,6i: 2 < i ≤m1}modd {1i, 4j, 5j, 62,6j: 2 ≤i≤m0,2< j ≤m3}meven 1162 {1i, 4j, 51,5j, 61,6j:m1< i ≤m, m3< j ≤m}modd {1i, 4i, 51,5j, 61,6i:m3< i ≤m, m4< i ≤m}meven 4151 {12,1i, 4i, 4m1,5i, 5m1,62,6i, 6m1: 2 < i < m1}modd {12,53,62}m= 4 {12,1i, 4i, 5i, 62,6i: 2 < i ≤m0}Other 4152 {1i, 4i, 5j, 6i, 6m:m1< i < m, m3< j ≤m}modd {1m3,1i, 4i, 5i, 5m, 6i, 6m:m3< i < m}meven 5161 {1m1,1i, 4i, 5i, 5m, 6i, 6m:m1< i < m}modd {1m3,1i, 4m3,4i, 5i, 5m, 6m3,6i, 6m:m3< i < m}meven 5261 {12,1i, 4i, 5j, 62,6i: 2 < i ≤m1,2< j ≤m3}modd {1i, 4j, 5j, 62,6j: 2 ≤i≤m0,2< j ≤m3,}meven 6162 {11,41,42,4m3,52,6m3}modd {11,1m3,41,42,52,5m4}meven 8 1161 {1i, 4i, 5i, 6i: 2 ≤i≤m1}modd {1i, 4j, 5j, 6j: 2 ≤i≤m0,2≤j≤m3}meven 1162 {1i, 1m, 41,4j, 51,5i, 61,6i:m1< i < m, m3< j ≤m}modd {1i, 41,4i, 51,5i, 61,6i:m3< i ≤m}meven 4151 {41,42,51,52}Cm= 2 {41,42,4(m−1),4m, 51,52,53,5m}Cm > 2 4152 {11,4m1}modd {11,1m0}meven 4161 {42,4m, 51,53} 5161 {41,4(m−1),52,5m} 6162 {11,6m3}modd {11,1m3}meven 9 5161 {41,4m, 51,61}C 10 1161 {1i, 4i, 5i, 6i: 2 ≤i≤m1}modd {1i, 4i, 5j, 6j: 2 ≤i≤m0,2≤j≤m3}meven 5161 ∅ 6162 {11,41,5m3,6m3}modd {11,1m3,41,4m3}meven 11 1161 {5m3}modd {4m3}meven 4151 {11,1m1,61}modd {11,61,6m0}meven 4161 {1i, 4j, 51,5j, 6i: 2 ≤i≤m1< j ≤m}modd {1i, 4k, 51,5l, 6j: 2 ≤i≤m0< k ≤m, 2≤j≤m3< l ≤m}meven 5161 {1m1,1i, 41,4j, 5j, 6i: 2 ≤j≤m1< i ≤m}modd {1i, 4j, 5k, 6i:m0< i ≤m, 1≤j≤m0,2≤k≤m3}meven Table 5: Resolving neighbourhoods of Gm(Hn), for 7 ≤n≤11. 178 A. Shahbaz, R. M. Falc´on and M. K. Mahmood n abcd RC{vb a, vd c}Case 12 1151 {12,1i, 42,4i, 52,5i, 61,6i: 2 < i ≤m1}modd {1i, 4i, 52,5j, 61,6j: 2 ≤i≤m0,2< j ≤m3}meven 1161 {1m1,1i, 41,51,4i, 4m, 5i, 5m, 6i:m1< i < m}modd {1i, 41,4i, 4m, 51,5j, 6k:m0< i < m, m4≤j≤m, m3< k < m}meven 4151 {1i, 4i, 5i, 6i: 2 ≤i≤m1≤i<m}modd {1i, 4i, 5j, 6j: 2 ≤i≤m0,2≤j≤m3}meven 5152 {11,41,5m3,6m3}modd {11,1m3,41,4m3}meven 5161 {11,1m, 62,6m} 13 1141 {1i, 4j, 5k, 61,6i: 2 ≤j≤m1< i ≤m, 2≤k≤m3}modd {1i, 4j, 5j, 61,6k:m0< i ≤m, 2≤j≤m3< j ≤m}meven 1161 {41,5m3}modd {41,4m3}meven 4151 {2m1,61}modd {61,6m3}meven 4161 {11,1i, 4j, 51,5j, 6i: 2 ≤i≤m1< j ≤m}modd {1i, 4j, 51,5k, 6l: 1 ≤i≤m0< j ≤m, 2≤l≤m3< k ≤m}meven 5161 {1m1,1i, 41,4j, 5j, 6i: 2 ≤j≤m1< i ≤m}modd {1i, 4j, 5k, 6i:m3≤i≤m, 1≤j≤m0,2≤k≤m3}meven 14 4161 {11,1m, 41,61}C 15 1161 {12,1i, 4i, 52,5i, 62,6i: 2 < i ≤m1}modd {1i, 4j, 5j, 6k: 2 ≤i≤m0,2< j ≤m3,2≤k≤m3}meven 5152 {11,1m1,1m, 41,5m3,61}modd {11,1m, 41,4m3,61,6m3}meven 5161 {1i, 4j, 5j, 6k:m1≤i < m, 2≤j≤m1< k ≤m}modd {1i, 4j, 5k, 6l: 2 ≤j≤m0< i < m, 2≤k≤m3≤l≤m}meven 6152 {1i, 4j, 51,5k, 6i: 2 ≤i≤m1< j ≤m, m3< k ≤m}modd {1i, 4j, 51,5m3,5j, 6k: 2 ≤i≤m0, m3< j ≤m, 2≤k≤m3}meven 16 1141 {11,41,61,62}C Table 6: Resolving neighbourhoods of Gm(Hn), for 12 ≤n≤16. Local fractional metric dimension of symmetric planar graphs 179 abcd RC{vb a, vd c}Case 1161 {1i, 1m1,41,4i, 5i, 5m1,6i, 6m1: 2 ≤i < m1}modd {1i, 41,4i, 5i, 6j: 2 ≤i≤m0,2≤j≤m3}meven 1162 {1i, 1m, 4i, 5i, 5m, 61,6i, 6m:m1< i < m}modd {1i, 1m, 4m3,4i, 5m3,5i, 5m, 61,6i, 6m:m3< i < m}meven 4151 {11,12,1i, 4i, 5i, 6j: 3 ≤i≤m1,2≤j≤m3}modd {11,12,1i, 4j, 5i, 62,6i: 3 ≤i≤m3,3≤j≤m0}meven 4152 {1i, 4j, 4m, 5j, 6i, 6m3: 1 ≤j≤m1≤i<m}Cmodd {1i, 4j, 4m, 5j, 6i, 6m4: 2 < i ≤m3,1≤j≤m3}Cmeven 4162 {42,4m, 51,52} 5161 {1i, 1m1,4i, 5i, 5m1,6i, 6m1,6m: 2 ≤i<m1}modd {1i, 4i, 5i, 6j, 6m, : 2 ≤i≤m0,2≤j≤m3}meven 5162 {1i, 1m, 41,4i, 5i, 5m, 61,6j:m1< i < m, m3< j ≤m}modd {1i, 1m, 41,4m3,4i, 5i, 5m, 61,6i, 6m:m3< i < m}meven 6162 {11,4m1,51,6m3}modd {11,1m3,51,5m3}meven Table 7: Resolving neighbourhoods of Gm(H17). GAsymptotic behaviour GAsymptotic behaviour Q1Unbounded H1, H2= 1 Q2Unbounded H3∼2 P1Unknown H4, H10, H12, H17 ∼2 P2∼2H5, H11, H13 ∼2 P3∼2H6, H9, H14, H16 Unknown P4, P5, P6∼2H7∼2 H8Unknown H15 ∼2 Table 8: Asymptotic behaviour of ldimf(Gm(G)).