EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6540 ISSN 1307-5543 – ejpam.com Published by New York Business Global Resolving Parameters in Generalized Sierpiński Networks Over Cycle of Length Five K. Bharani Dharan1, S. Radha1,∗ 1 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai, Tamil Nadu, India 600127 Abstract. The fascinating family of fractal-based networks known as generalized Sierpiński net- works has garnered significant interest across various research domains and practical applications. These graphs exhibit self-similar structures, making them particularly valuable in areas such as antenna structures, interconnection networks and the porous materials. Their recursive nature and hierarchical organization enhance their relevance in modeling complex systems and network structures. A key parameter for analyzing these graphs is the metric dimension, which represents the smallest set of reference vertices (or landmarks) needed to uniquely identify the distances be- tween all other vertices in the graph. This parameter is crucial for network localization, efficient routing, and information retrieval. Beyond the standard metric dimension, other variations play important roles in different applications. The fault-tolerant metric dimension is essential in ro- bust network design, ensuring that localization remains possible even if certain reference points fail. The edge metric dimension is widely used in network security and surveillance, where mon- itoring specific connections is more relevant than individual nodes. Meanwhile, the fault-tolerant edge metric dimension has applications in resilient communication networks, guaranteeing reliable identification of edges even under failure conditions. In this study, we specifically examine the metric, fault-tolerant metric, edge metric, and fault-tolerant edge metric dimensions of generalized Sierpiński networks over C5. These findings provide deeper insights into their structural properties and distinguish them from traditional cycle networks, highlighting their potential in real-world applications. 2020 Mathematics Subject Classifications: 05C12, 94C15, 68M10 Key Words and Phrases: Generalized Sierpiński network; metric dimension; fault-tolerant metric dimension; edge metric dimension; fault-tolerant edge metric dimension 1. Introduction Graph theory plays a crucial role in distributed parallel computing by providing a math- ematical framework for modeling, analyzing, and optimizing system performance [1]. In ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6540 Email addresses: bharanidharan.k2022@vitstudent.ac.in (K. Bharani Dharan), radha.s@vit.ac.in (S. Radha) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) K. B. Dharan, S. Radha / Eur. J. Pure Appl. Math, 18 (3) (2025), 6540 2 of 17 such systems, multiple processors collaborate to execute tasks efficiently, requiring well- structured communication and coordination, which can be effectively represented using graphs [2]. Various graph-theoretic techniques enhance different aspects of parallel com- puting, such as task scheduling, load balancing, fault tolerance, and network optimiza- tion [3]. Graph partitioning helps distribute workloads evenly, preventing bottlenecks, while interconnection network models, including hypercubes, meshes, and trees, assist in minimizing communication latency [4]. Connectivity and domination properties con- tribute to fault tolerance by ensuring alternative paths in case of node or link failures [5]. Graph-based methods like similarity and clustering also enhance security through anomaly detection in network traffic, helping mitigate unauthorized access and cyber threats [6]. Additionally, shortest path algorithms improve data transmission efficiency, reducing congestion in communication networks [7]. An effective approach particularly in designing scalable and robust parallel computing networks involves fractal-structured graphs, such as generalized Sierpiński networks and fractal cubic networks which exhibit self-similarity and hierarchical organization [8, 9]. These graphs offer several advantages, including seamless scalability, as their recursive nature allows for structured expansion without compromising efficiency [10]. Their hierarchical architecture enables optimized routing and communication, reducing congestion and improving data flow [11]. Moreover, their inherent redundancy enhances fault tolerance, ensuring system resilience even in the event of node or link failures. The structured nature of fractal graphs also facilitates balanced workload distribution, improving computational efficiency while simultaneously reducing energy consumption in communication, making them ideal for energy-efficient parallel computing architectures [12]. These networks have applications in various artifi- cial and natural systems, including neuroscience [13], computer networks and musics [14]. They also assist in analyzing complex biological structures like bacterial growth patterns [15]. Among these structures, Sierpiński networks are particularly relevant to parallel computing, especially in designing high-performance computing clusters and supercom- puters. Overall, graph theory, particularly through fractal-based structures, significantly contributes to optimizing distributed parallel computing by enhancing performance, scala- bility, security, and reliability, making it a key tool in high-performance computing system design [16]. The Sierpiński network is distinguished by its recursive hierarchical arrangement, where each level represents a smaller version of the overall structure. Because of these charac- teristics, they are seen to be a viable topology for systems that use parallel computing, especially in fields of study that are looking into new network architectures to improve fault tolerance, scalability, and efficiency [8]. Tower of Hanoi graphs with three pegs are a particular type of Sierpiński networks that occur when the base graph is a complete graph K3. Recursive networks are created by extending Sierpiński graphs by appending an open connection to their extreme vertices. These networks have been used in Very Large Scale Integration (VLSI) designs since their initial introduction in 1988 as a foundation for message-passing architectures [17]. Sierpiński graphs were created in the late 1990s when Sierpiński labelings were applied to WK-recursive networks [18]. Deeper investigation of their characteristics has been made K. B. Dharan, S. Radha / Eur. J. Pure Appl. Math, 18 (3) (2025), 6540 3 of 17 possible by this labelling strategy. In [19], a number of significant features of Sierpiński graphs have been discussed. These networks have been shown to have Hamiltonian characteristics and their geodesic distances between vertices have been calculated [20]. Additional metrics have been examined, such as median values, average eccentricity, and connectedness [21–23]. Furthermore, several Sierpiński graph topological descriptors have been studied [24], which has shed light on the shortest path structures in these networks [25]. Complete graphs serve as the basic building blocks for the construction of classical Sierpiński networks, as described in [18]. In [19], a more expansive category called as generalised Sierpiński networks was established and given the survey about their various graph theo- retical properties. The concept of the Sierpiński product G⊗fH, where f : V (G) → V (H), generalizes the construction of Sierpiński-type graphs by embedding copies of H based on the structure of G. This construction and related metric properties have been rigorously studied in [26]. The study of metric dimension in recursive and fractal-based networks has drawn increasing attention due to its implications in network navigation and infor- mation retrieval [27–32]. In particular, the work presented in [33] investigates the metric dimension and its variants for generalized Sierpiński networks constructed over C4, which are known to be spanning subgraphs of hypercubes. Their analysis showcases the inter- play between structural properties, such as vertex twins, and the complexity of resolving sets. Motivated by these findings, we extend the investigation to the class of generalized Sierpiński networks over C5, exploring how the odd cycle structure influences the metric dimension and its fault-tolerant variants. 2. Preliminaries Let Γ = (V(Γ),E(Γ)) be a graph of order n. For h ∈ N, the generalized Sierpiński graph [19], denoted as Sh Γ, has the vertex set V(Sh Γ) = V(Γ)h. The notation for a vertex g = (g1, g2, . . . , gh) in V(Sh Γ) is abbreviated as g = g1g2 . . . gh. Two vertices, f = f1f2 . . . fh and g = g1g2 . . . gh, are adjacent if there exists an index x ∈ Zh+1 \ {0} such that for all y ∈ Zh+1 \ {0}, the following properties hold: i) If y < x, then fy = gy; ii) If y = x, then fx ̸= gx and fxgx ∈ E(G); iii) If y > x, then fy = gx and fy = gx. For example, the graphs S1 K4 and S2 K4 are depicted in Figure 1. The graph Sh Γ can be constructed recursively from a base graph Γ as follows. Let S1 Γ = Γ. For each h ≥ 2, consider n disjoint copies of the graph Sh−1 Γ . In the xth copy, where x ∈ Zn, prefix each vertex label with the index x. These subgraphs are denoted by xSh−1 Γ , for all x ∈ Zn. Furthermore, if two vertices f and g are adjacent in Γ, then in Sh Γ, the vertex labeled fgh−1 is adjacent to the vertex labeled gfh−1. Generalized Sierpiński networks have been used to represent polymer networks [34]. Roman domination [35], strong metric dimension [36], and other graph-theoretic features K. B. Dharan, S. Radha / Eur. J. Pure Appl. Math, 18 (3) (2025), 6540 4 of 17 (a) (b) Figure 1: (a)S1 K4 ∼= K4 (b)S2 K4 such as chromatic number, vertex cover number and clique number for Sh G are already determined. The topological indices of generalized Sierpiński networks are examined in [34]. In this work, we particularly analyze the family of Sh C5 and calculate its metric, fault-tolerant metric, edge metric and fault-tolerant edge metric dimensions. The graphs S1 C5 , S2 C5 and S3 C5 are listed in the Figure 2. 0 1 23 4 (a) 00 01 0203 04 10 11 12 13 14 20 2124 22 23 30 31 32 33 34 40 41 42 43 44 (b) 000 001 002003 004 010 011 012013 014 020 021 000 001 002003 004 010 011 012013 014 022 023 024 030 031 032 033 034 040 041 042 043 044 120 121 100 101 102103 104 110 111 112113 114 122 124 130 131 132 134 140 141 142143 144 420 421 400 401 402403 404 410 411 412 413 414 422 424 430 431 432 434 440 441 442 443 444 320 321 300 301 302 303 304 310 311 312313 314 322 324 330 332 332 334 340 341 342 343 344 220 221 200 201 202203 204 210 211 212 213 214 222 224 230 231 232 234 240 241 242 243 244 123 133 223233323333 423 433 (c) Figure 2: (a)S1 C5 (b)S2 C5 (c)S3 C5 K. B. Dharan, S. Radha / Eur. J. Pure Appl. Math, 18 (3) (2025), 6540 5 of 17 2.1. Metric basis and fault-tolerant basis Networks are considered to be graphs with intriguing properties based on graph theory. The effectiveness of locating and differentiating each vertex of a graph using a small num- ber of landmarks is measured by its metric dimension [37–39]. Location-based services, robotics and network design are just a few of the domains where metric dimension finds use [40, 41]. It aids in determining the best locations for sensors inside a network to guarantee accurate location identification [42]. Determining an exact value of this parameter of a graph is difficult task. There are effective methods for calculating this parameter for some graph types like trees. The problem of determining this parameter for directed graphs is NP-hard [29], bipartite graphs[43] and general graphs [37]. Despite the computational difficulties, this parameter is calculated for many graph networks, such as honeycomb [44], butterfly [43], Beneš [43], circulant graphs [45], generalized subdivision of prism [46], chemical structures [47], convex triangular networks [48], multistage interconnection net- works [49] and Sierpiński [50]. A current assessment of the literature on metric dimension could be found in [51]. The distance dΓ(f, g) in a connected graph Γ is defined as the smallest number of edges re- quired to travel from a vertex f to a vertex g. For a vertex u and an edge e = fg (with f, g ∈ V(Γ)), the distance from u to the edge e is defined as dΓ(u, e) = min{dΓ(u, f), dΓ(u, g)}. Given an ordered subset Y = {y1, y2, . . . , yℓ} ⊆ V(Γ), the representation of a vertex v ∈ V(Γ) with respect to Y is the ℓ-tuple r(v|Y) = (dΓ(v, y1), dΓ(v, y2), . . . , dΓ(v, yℓ)). A subset Y is called a resolving set if each vertex in Γ has a unique representation con- cerning Y. The metric dimension of Γ, denoted by dim(Γ), is the minimum cardinality among all resolving sets in Γ. For the Sierpiński network S2 C5 , the two vertex subset W = {00, 20} is enough to make other vertices to be resolved. We can easily verify that the representation of each vertex with respect to the set {00, 20} are distint in the Figure 3 and the cardinality of resolving set cannot be 1 for S2 C5 as it is not isomorphic to a path. Then the set W is a metric basis for S2 C5 . This implies dim(S2 C5 ) = 2. (0,6) (1,5) (2,6) (2,7) (1,7) (2,4) (3,3) (4,2) (4,3) (3,4) (6,0) (5,1)(7,1) (6,2) (7,2) (6,5) (7,4) (7,3) (6,4) (5,5) (2,8) (3,7) (3,8) (4,7) (4,6) Figure 3: Metric representation of each vertex with respect to the set W = {00, 20} in S2 C5 Many variations have been developed from this well-established concept in the litera- K. B. Dharan, S. Radha / Eur. J. Pure Appl. Math, 18 (3) (2025), 6540 6 of 17 ture [18, 52–56]. Fault-tolerant metric dimension is one of these new and highly motivated variants. The key idea in this version is that even if one vertex in the chosen set of vertices becomes flawed or unusable, the graph must still be resolved using that set. Let F ⊆ V(Γ) is called a fault-tolerant resolving set, if we have r(u|F \ {x}) ̸= r(v|F \ {x}), for every x ∈ F and u, v ∈ V(Γ)(for every pair of distinct vertices u, v ∈ V(Γ), there exist at least two vertices x, y ∈ F such that dΓ(u, x) ̸= dΓ(v, x) and dΓ(u, y) ̸= dΓ(v, y)). Among all such sets, one with the smallest possible size is called a fault-tolerant metric basis. The number of vertices in a fault-tolerant metric basis is known as the fault-tolerant metric dimension of Γ, and it is denoted by dim′(Γ). This idea was first presented in [57], and was further discussed in [5, 45, 58]. 2.2. Edge metric and fault-tolerant edge metric basis In parallel computing systems, an interconnection network comprising a structured con- figuration of processors and communication links, plays a vital role in facilitating data exchange among processors. Efficient fault detection within such networks requires the ability to distinguish between communication links. This can be accomplished by identi- fying a minimal set of vertices capable of uniquely determining every edge in the network graph. A subset S ⊆ V(Γ) is called an edge resolving set if, for every pair of distinct edges e1, e2 ∈ E(Γ), there exists a vertex w ∈ S such that dΓ(w, e1) ̸= dΓ(w, e2). The smallest possible size of such a set is defined as the edge metric dimension of Γ, denoted by dimE(Γ). Kelenc et al. [52] were the first to study this metric dimension variant and proved its NP- completeness. Since then, numerous research articles have explored this topic, including studies on the dimension of convex polytope graphs [59], web graphs, prism-related graphs, generalized Petersen graphs [60] and silicate networks [61]. Other works have examined this parameter for Erdős–Rényi random graphs [62], certain classes of planar graphs [63], and the identification of graph network with higher dimension [64, 65]. In [66], the metric and edge metric dimension of hypercubes were analyzed. Additionally, researchers have investigated graph operations such as join of graph networks, lexicographic product and corona product [67], as well as hierarchical products across various graph classes [68]. Recently, increasing attention has been given to identifying graphs where dimE < dim [69, 70]. A subset F ⊆ V(Γ) is said to be a fault-tolerant edge resolving set if, for every vertex u ∈ F , the set F \ {u} remains an edge resolving set of Γ. Among all such subsets, one with the smallest possible size is called a fault-tolerant edge metric basis. The cardinality of a fault-tolerant edge metric basis is referred to as the fault-tolerant edge metric dimension of Γ, denoted by dimE′(Γ). K. B. Dharan, S. Radha / Eur. J. Pure Appl. Math, 18 (3) (2025), 6540 7 of 17 Figure 4: Edge representation of each edge with respect to the set W = {02, 20} in S2 C5 From Figure 4, we can easily conclude that dimE(S 2 C5 ) = 2, and it is easy to verify that the fault-tolerant edge metric dimension of S2 C5 is equal to 4, where {02, 20, 03, 30} is the fault- tolerant metric basis. In this paper, we use the following notations to represent various parameters: RS for metric resolving set, MB for metric basis, dim for metric dimension, FTRS for fault-tolerant metric resolving set, FTMB for fault-tolerant metric basis, dim′ for fault-tolerant metric dimension, ERS for edge resolving set, EMB for edge metric basis, dimE for edge metric dimension, FTERS for fault-tolerant edge resolving set, FTEMB for fault-tolerant edge metric basis, and dim′ E for fault-tolerant edge metric dimension. 3. Main results In this section, we present results pertaining to the exact values of both the dim(Γ) and the dim′(Γ)s, as outlined in Subsection 3.1. Subsequently, Subsection 3.2 addresses the corresponding results for the dimE(Γ) and the dim′ E(Γ). To support the upcoming theorems, we introduce a particular subset of vertices that will be instrumental in the analysis. Let Γ′ be an induced subgraph of Sh C5 , and let W denote a resolving set for Sh C5 with h ≥ 3. A vertex v ∈ V(Γ′) is referred to as a pseudo resolver for Γ′ if there exists a vertex u ∈ W \ V(Γ′) such that, ∀x ∈ V(Γ′), the equality dΓ(x, u) = dΓ(x, v) + dΓ(v, u) holds. This implies that replacing u in W with v results in a set that still resolves Γ′, that is, r(x|(W \ {u}) ∪ {v}) ̸= r(y|(W \ {u}) ∪ {v}) for all distinct x, y ∈ V(Γ′). Figure 5 illustrates the collection of pseudo resolver vertices for each induced subgraph iS2 C5 , where i ∈ Z5, within the graph S3 C5 . K. B. Dharan, S. Radha / Eur. J. Pure Appl. Math, 18 (3) (2025), 6540 8 of 17 000 001 002003 004 010 011 012013 014 020 021 000 001 002003 004 010 011 012013 014 022 023 024 030 031 032 033 034 040 041 042 043 044 120 121 100 101 102103 104 110 111 112113 114 122 124 130 131 132 134 140 141 142143 144 420 421 400 401 402403 404 410 411 412 413 414 422 424 430 431 432 434 440 441 442 443 444 320 321 300 301 302 303 304 310 311 312313 314 322 324 330 332 332 334 340 341 342 343 344 220 221 200 201 202 203 204 210 211 212213 214 222 224 230 231 232 234 240 241 242 243 244 123 133 223233323333 423 433 Figure 5: S3 C5 with vertices of the resolving set encircled with green color and pseudo resolvers for each S2 C5 encircled with violet color 3.1. Metric dimension of generalized Sierpiński networks over cycle of length five We can easily say that dim(S1 Cn ) = 2 as S1 Cn is isomorphic to Cn. Then, dim(S2 C5 ) = 2 from the Section 2.1. Consider the graph S3 C5 . It has exactly 5 induced subgraphs as S2 C5 the dim of those induced subgraphs is 2. Then let us consider the subset W = {[x]3 : x ∈ Z5} of S3 C5 . Then for each induced subgraph S2 C5 there are exactly two pseudo resolvers in the subset {i1i22 : i2 ≡ (i1± 1)mod 5} ⊂ V(i1S2 C5 ). These pseudo resolvers serve a crucial role: although they are not part of W , but due to their property (given in definition of pseudo resolvers), their relative positions with respect to the remaining vertices in W allow them to replace the resolvers that would belong to the subgraph i1S 2 C5 . This implies that the every pseudo resolvers of i2S 2 C5 along with the vertices in i1S 2 C5 ∩W are enough to resolve all the vertices of i1S 2 C5 , i1 ∈ Z5. This implies W is enough to resolve all the vertices of S3 C5 . So that, dim(S3 C5 ) ≤ 5. Suppose that dim(S3 C5 ) = 4. By the pigeonhole principle, there exist at least one induced subgraph S2 C5 that do not contain any vertex from W . Then there exists at least two vertices that have the same representation, a contradiction. This implies that dim(S3 C5 ) ≥ 5. Then dim(S3 C5 ) = 5. The following theorem gives the K. B. Dharan, S. Radha / Eur. J. Pure Appl. Math, 18 (3) (2025), 6540 9 of 17 value of dim(Sh C5 ) for any h ≥ 3. Theorem 1. If h ≥ 3, then dim(Sh C5 ) = 5(1+5h−3) 2 . Proof. Let W3 = {x3 : x ∈ Z5} and for h ≥ 4, Wh = 4⋃ i=0 iWh−1 \ {i((i− 1)mod 5)h−1, i((i+ 1)mod 5)h−1}. We claim that Wh is a RS of Sh C5 , h ≥ 4 and proceed by induction on h. Already we know that the claim is true for h = 3. So it remains to verify that, Wh+1 is RS of Sh+1 C5 , h ≥ 4. Consider any two arbitrary vertices u = u1u2 . . . uh+1 and v = v1v2 . . . vh+1 of Sh+1 C5 . Case 1: u1 − v1 ≡ ±1 mod 5 Suppose there exists some vertices x ∈ u1S h C5 and y ∈ v1S h C5 such that r(x|Wh+1 ∩ u1S h C5 ) = r(y|Wh+1 ∩ u1S h C5 ) but there exists t ∈ Wh+1 ∩ ((v1 + 1) mod 5)Sh C5 such that dSh+1 C5 (x, t) ̸= dSh+1 C5 (y, t) when u1 < v1 or there exists t ∈ Wh+1∩ ((u1+1) mod 5)Sh C5 such that dSh+1 C5 (x, t) ̸= dSh+1 C5 (y, t) when u1 > v1. This implies that r(x|Wh+1) ̸= r(y|Wh+1) for all x ∈ u1S h C5 and y ∈ v1S h C5 . Then u and v have distinct representation with respect to Wh+1 when u1 − v1 ≡ ±1(mod 5). Case 2: u1 − v1 ̸≡ ±1(mod 5) In this case, we have dSh+1 C5 (u, t) < dSh+1 C5 (v, t) for all t ∈ Wh+1 ∩ u1S h C5 . Thus, u and v have distinct representation with respect to Wh+1 when u1 < v1 and u1 − v1 ̸≡ 1(mod 5). Case 3: u1 = v1 We may assume that u1 = v1 = 0. By induction and the structure of Sh+1 C5 , the set 0Wh resolves all the vertices in V (0Sh C5 ). By the construction, 01h, 04h ∈ 0Wh but these ver- tices becomes pseudo resolvers for 0Sh C5 as for every x ∈ 0Sh C5 , we have dSh+1 C5 (x, t) = dSh+1 C5 (x, 01h) + dSh+1 C5 (01h, t), for t ∈ Wh+1 ∩ 2Sh C5 and dSh+1 C5 (x, t) = dSh+1 C5 (x, 04h) + dSh+1 C5 (04h, t), for t ∈ Wh+1∩3Sh C5 . This implies that r(u|W h+1) ̸= r(v|W h+1) for u ̸= v and u1 = v1. Thus, we conclude that Wh+1 is a RS of Sh+1 C5 . For h ≥ 4, |Wh| = 5(|Wh−1| − 2), where |W3| = 5. Solving this recurrence relation, we get |Wh| = 5(1+5h−3) 2 for h ≥ 3. Thus dim(Sh C5 ) ≤ 5(1+5h−3) 2 . To prove that dim(Sh C5 ) ≥ 5(1+5h−3) 2 , let us assume that there exist a resolving set W , such that |W | = 5(1+5h−3) 2 − 1. Then by pigeonhole principle, there exist at least one induced subgraph i1i2 . . . ih−2S 2 C5 with only two pseudo resolvers u, v ∈ i1i2 . . . ih−2S 2 C5 such that uh = uh−1 = uh−2 ± 1 (mod 5) and V(i1i2 . . . ih−2S 2 C5 ) ∩W = ∅. Then we have either r(i1 . . . ih−2(ih−2 − 1)(ih−2 + 2)|W ) = r(i1 . . . ih−2(ih−2 − 2)(ih−2 − 1)|W ) or r(i1 . . . ih−2(ih−2 + 1)(ih−2 − 2)|W ) = r(i1 . . . ih−2(ih−2 + 2)(ih−2 + 1)|W ), K. B. Dharan, S. Radha / Eur. J. Pure Appl. Math, 18 (3) (2025), 6540 10 of 17 a contradiction. This leads to dim(Sh C5 ) ≥ 5(1+5h−3) 2 . Hence, dim(Sh C5 ) = 5(1+5h−3) 2 for h ≥ 3. Lemma 1. Let R3 = {x((x+ 2)mod 5)2 : x ∈ Z5}. For h ≥ 4, Rh = ⋃4 i=0 iRh−1 \ {i((i+ 1)mod 5)h−2((i+ 3)mod 5), i((i− 1)mod 5)h−2((i− 3)mod 5)} is a RS for Sh C5 . Proof. The case h = 3 can be verified directly. Assume, as the induction hypothesis, that the claim holds for h ≥ 4. We proceed to verify the claim for h + 1. Let p = p1p2 . . . ph+1 and q = q1q2 . . . qh+1 be two arbitrary vertices of Sh+1 C5 . Case 1: p1 − q1 ≡ ±1 mod 5 Suppose there exists some vertices x ∈ p1S h C5 and y ∈ q1S h C5 such that r(x|Rh+1∩p1Sh C5 ) = r(y|Rh+1 ∩ p1S h C5 ) but there exists t ∈ Rh+1 ∩ ((q1 + 1) mod 5)Sh C5 such that dSh C5 (x, t) ̸= dSh C5 (y, t) when p1 < q1 or there exists t ∈ Rh+1∩((p1+1) mod 5)Sh C5 such that dSh C5 (x, t) ̸= dSh C5 (y, t) when p1 > q1. This implies that r(x|Rh+1) ̸= r(y|Rh+1) for all x ∈ p1S h C5 and y ∈ q1S h C5 . Then p and q have distinct representation with respect to Rh+1 when p1 − q1 ≡ ±1(mod 5). Case 2: p1 − q1 ̸≡ ±1(mod 5) In this case, we have dSh+1 C5 (p, t) < dSh+1 C5 (q, t) for all t ∈ Rh+1 ∩ p1S h C5 . So p and q have distinct representation with respect to Rh+1 when p1 < q1 and p1 − q1 ̸≡ 1(mod 5). Case 3: p1 = q1 Now assume that p1 = q1. We may assume that p1 = q1 = 0. By induction and the structure of Sh+1 C5 , the set 0Rh resolves all the vertices in V (0Sh C5 ). By the construction, 01h, 04h ∈ 0Rh but these vertices becomes pseudo resolvers for 0Sh C5 as for every x ∈ 0Sh C5 , we have dSh+1 C5 (x, t) = dSh+1 C5 (x, 01h) + dSh+1 C5 (01h, t), for t ∈ Rh+1 ∩ 2Sh C5 and dSh+1 C5 (x, t) = dSh+1 C5 (x, 04h)+dSh+1 C5 (04h, t), for t ∈ Rh+1∩3Sh C5 . This implies that r(p|Rh+1) ̸= r(q|Rh+1) for p ̸= q and p1 = q1. Thus, we conclude that Rh+1 is a RS of Sh+1 C5 . To prove the dim′ of Sh C5 for h ≥ 3, we defined a MB Rh in the Lemma 1, such that Rh ∩ Wh = ∅. Then W ′ h = Rh ∪ Wh is also a RS and W ′ h is obviously a FTRS for Sh C5 , when h ≥ 3 and the proof for W ′ h to be a FTB is given in the following theorem. Theorem 2. For h ≥ 3, dim′(Sh C5 ) = 5(1 + 5h−3). Proof. We know that W ′ h = Rh ∪ Wh is a FTRS for Sh C5 , when h ≥ 3. This implies that dim′(Sh C5 ) ≤ 5(1 + 5h−3), for h ≥ 3. Now, we need to prove that dim′(Sh C5 ) ≥ 5(1 + 5h−3). Suppose that there exist a FTRS W ′, such that |W ′| = 5(1 + 5h−3) − 1. Then by pigeonhole principle, there exists i1, i2, . . . , ih−2 ∈ Z5, such that |i1i2 . . . ih−2S 2 C5 ∩ W ′| = 1 with two pseudo re- solvers, that is not sufficient to fault-tolerantly resolve all the vertices of the induced subgraph i1i2 . . . ih−2S 2 C5 , a contradiction. So, for h ≥ 3, dim′(S2 Ch ) ≥ 5(1+5h−3), implies dim′(S2 Ch ) = 5(1 + 5h−3). K. B. Dharan, S. Radha / Eur. J. Pure Appl. Math, 18 (3) (2025), 6540 11 of 17 3.2. Edge metric dimension of generalized Sierpiński networks over cycle of length five We can easilty say that dimE(S 1 C5 ) = 2 as S1 Cn is isomorphic to Cn. For S2 C5 , let W2 = {02, 20} be the subset of V(S2 C5 ). From Figure 4, we can easily verify that each edges having unique representation with respect to W2 and dimE(S 2 C5 ) cannot be less than 2 as it is not isomorphic to path graph. For S3 C5 , let W3 = {022, 133, 244, 300, 411} be the subset of V (S3 C5 ). Then each edge in- cident to V(0S2 C5 ) is resolved by the vertices in {022, 133, 411} where the vertices 011, 044 acts as an pseudo resolver for the edges incident with V(0S2 C5 ). Now, suppose that there ex- ists two edges etu (t, u ∈ V(0S2 C5 )) and evw (v ∈ V(iS2 C5 ) or w ∈ V(iS2 C5 ), i > 0), such that dS3 C5 (etu, 022) = dS3 C5 (evw, 022), then we have dS3 C5 (etu, i(i+2)2) ≥ dS3 C5 (evw, i(i+2)2)+1. This implies that edges incident to the vertices in V(0S2 C5 ) have distinct representation in E(S3 C5 ). Thus, dimE(S 3 C5 ) ≤ 5. Due to symmetrical property, all the edges in S3 C5 have unique representation with respect to W3. This implies dimE(S 3 C5 ) ≤ 5. Suppose that there exists an ERS U with cardinality 4. Then by pigeonhole principle, there exists i ∈ Z5, such that V(iS2 C5 ) ∩ U = ∅. Then there exists two edges etu and evw (t = i((i + 2)mod 5)2, u = i((i + 2)mod 5)((i + 1)mod 5), v = i((i + 1)mod 5)((i − 2)mod 5) and w = i((i + 1)mod 5)((i − 1)mod 5)) in an induced subgraph iS2 C5 , whose edge representations with respect to U are same, which is a contradiction and implies that dim(S3 C5 ) ≥ 5. Hence dim(S3 C5 ) = 5. Theorem 3. For h ≥ 3, dimE(S h C5 ) = 5h−2. Proof. Let W3 = {x(x+ 2)2 : x ∈ Z5} and for h ≥ 4, let Wh = 4⋃ i=0 iWh−1 We claim that Wh is a ERS for Sh C5 , h ≥ 4 and prove this by induction on h. We know that the claim is true for h = 3. Thus, it needs to verify that Wh+1, h ≥ 4, is a ERS of Sh+1 C5 . For any etu, evw ∈ E(Sh+1 C5 ), the following cases will occur. Case 1: t, u, v, w ∈ V(0Sh C5 ) By the construction of Sh+1 C5 and ERSWh+1, we have r(etu|Wh+1∩V (0Sh C5 )) ̸= r(evw|Wh+1∩ V (0Sh C5 )) for all t, u, v, w ∈ V(0Sh C5 ). This implies that no two edges incident only with the vertices in V(0Sh C5 ) have same edge representation. Case 2: t, u ∈ V(0Sh C5 ) and v, w ̸∈ V(0Sh C5 ) In this case, there exists atleast one vertex x ∈ Wh+1 ∪ 0Sh C5 such that dSh+1 C5 (etu, x) < dSh+1 C5 (evw, x). This implies that r(etu|Wh+1) ̸= r(evw|Wh+1) for t, u ∈ V(0Sh C5 ) and v, w ̸∈ V(0Sh C5 ). Case 3: t, u, v ∈ V(0Sh C5 ) and w ̸∈ V(0Sh C5 ) In this case, there exists some edges etu such that r(etu|Wh+1∪0Sh C5 ) = r(evw|Wh+1∪0Sh C5 ). But, there exists some x ∈ Wh+1 \ V(0Sh+1 C5 ), we have dSh+1 C5 (etu, x) > dSh+1 C5 (evw, x). K. B. Dharan, S. Radha / Eur. J. Pure Appl. Math, 18 (3) (2025), 6540 12 of 17 From the above cases, we can say that every edge incident to V(0Sh C5 ) have distinct representations. Due to the symmetrical structure of Sh+1 C5 , all the edges in Sh+1 C5 have distinct edge representations. This implies that Wh+1 is an ERS of Sh+1 C5 . So that for h ≥ 4, |Wh| = 5(|Wh−1|). Solving this recurrence relation, we get |Wh| = 5h−2 for h ≥ 4, which implies that dimE(S h C5 ) ≤ 5h−2. Now to prove that dimE(S h C5 ) ≥ 5h−2, let us suppose that there exist a resolving set W with cardinality less that 5h−2. Let us assume that dimE(S h C5 ) = 5h−2 − 1. Then by pigeonhole principle, there exist at least one induced subgraph i1i2 . . . ih−2S 2 C5 (i1, i2, . . . , ih−2 ∈ Z5) with only two pseudo resolvers u, v ∈ i1i2 . . . ih−2S 2 C5 such that uh = uh−1 = uh−2 ± 1 and V(i1i2 . . . ih−2S 2 C5 ) ∩ W = ∅. Then there exists two edges etu, evw ∈ E(S2 C5 ) (t = i1 . . . ih−2((ih−2+2)mod 5)2, u = i1 . . . ih−2((ih−2+2)mod 5)((ih−2+ 1)mod 5), v = i1 . . . ih−2((ih−2 + 1)mod 5)((ih−2 − 2)mod 5) and w = i1 . . . ih−2((ih−2 + 1)mod 5)((ih−2−1)mod 5)) have the same representation with respect to W , a contradic- tion. This leads to dimE(S h C5 ) ≥ 5h−2. Hence, dimE(S h C5 ) = 5h−2. Lemma 2. Let U3 = {033, 144, 200, 311, 422}. For h ≥ 4, Uh = ⋃4 i=0 iUh−1 is an EMB for Sh C5 . The proof of the above lemma is similar to the proof of Theorem 3, due to the reflexive property of Sh C5 . To prove the dimE′ of Sh C5 for h ≥ 3, we defined a EMB Uh in the Lemma 2, such that Uh ∩Wh = ∅. Then W ′ h = Uh ∪Wh is also an ERS and W ′ h is obviously a FTERS for Sh C5 , when h ≥ 3 and the proof for W ′ h to be a FTEB is given in the following theorem. Theorem 4. For h ≥ 3, dimE′(Sh C5 ) = 2 · 5h−2. Proof. Let U3 and W3 be the distinct ERS of S3 C5 , such that U3 ∩W3 = ∅. Obviously, the set U3 ∪ W3 will be the FTERS for S3 C5 , proving that dimE′(S3 C5 ) ≤ 10. To prove the sufficient part, let us assume that there exist a FTERS S, such that |S| < 10. Let us assume that |S| = 9. Then, by pigeonhole principle, there exist i ∈ Z5 , such that V(iS2 C5 ) ∩ W ′ < 2 with only two pseudo resolvers {i(i + 1)2, i(i − 1)2}. When removing that one vertex in V(iS2 C5 ) ∩ S, there exists two edges incident to the vertices in V(iS2 C5 ) that have the same edge representation, which implies that any subset W ′ with cardinality less than 10 cannot be a FTERS for S3 C5 . For h ≥ 4, let W ′ h = Wh ∪ Uh, where Wh = ⋃4 i=0 iWh−1 and Uh = ⋃4 i=0 iUh−1. We know that Wh and Uh are the ERS for Sh C5 for any h ≥ 2 and Wh ∩ Uh = ∅. Then W ′ h be the FTERS for Sh C5 , ∀h ≥ 4. We have |W ′ h| = 5 · |W ′ h−1|. Solving this recurrence relation, we get |W ′ h| = 2 · 5h−2. This implies that dimE′(Sh C5 ) ≤ 2 · 5h−2. Now we need to prove that dimE′(Sh C5 ) ≥ 2·5h−1. Let us assume the contrary that there exist a FTERSW ′ h, such that |W ′ h| = 2·5h−2−1. Then, there exist i1, i2, . . . , ih−2 ∈ Z5 such K. B. Dharan, S. Radha / Eur. J. Pure Appl. Math, 18 (3) (2025), 6540 13 of 17 that V(i1i2 . . . ih−2S 2 C5 )∩W ′ h < 2 with two pseudo resolvers i1i2 . . . ih−2((ih−2+1)mod 5)2 and i1i2 . . . ih−2((ih−2−1)mod 5)2, where these vertices are not enough to fault-tolerantly resolve the edges incident to the vertices in V(i1i2 . . . ih−2S 2 C5 ), a contradiction. So, we conclude that dimE′(Sh C5 ) ≥ 2 · 5h−2, ∀h ≥ 3. Then dimE′(Sh C5 ) = 2 · 5h−2. 4. Conclusion In this study, we analyzed dim, dim′, dimE, and dimE′ of generalized Sierpiński net- works over C5. Our findings reveal that the fault-tolerant (edge) metric dimension is directly proportional to the (edge) metric dimension with proportionality constant 2, for this graph family. This relationship highlights the structural consistency of these graphs and provides a fundamental insight into their resolvability and robustness in network applications. The proportionality observed in our results has practical significance in fault-tolerant network design, where ensuring efficient localization despite failures is cru- cial. Additionally, these findings contribute to a deeper understanding of fractal-based graph structures, reinforcing their applicability in areas such as routing, surveillance, and resilient communication systems. 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