Fault-Tolerant Metric Dimension of Two-Fold Heptagonal-Nonagonal Circular Ladder

Let [Formula: see text] be an undirected (i.e., all the edges are bidirectional), simple (i.e., no loops and multiple edges are allowed), and connected (i.e., between every pair of nodes, there exists a path) graph. Let [Formula: see text] denotes the number of edges in the shortest path or geodesic distance between two vertices [Formula: see text]. The metric dimension (or the location number) of some families of plane graphs have been obtained in [M. Imran, S. A. Bokhary and A. Q. Baig, Families of rotationally-symmetric plane graphs with constant metric dimension, Southeast Asian Bull. Math. 36 (2012) 663–675] and an open problem regarding these graphs was raised that: Characterize those families of plane graphs [Formula: see text] which are obtained from the graph [Formula: see text] by adding new edges in [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, by answering this problem, we characterize some families of plane graphs [Formula: see text], which possesses the radial symmetry and has a constant metric dimension. We also prove that some families of plane graphs which are obtained from the plane graphs, [Formula: see text] by the addition of new edges in [Formula: see text] have the same metric dimension and vertices set as [Formula: see text], and only 3 nodes appropriately selected are sufficient to resolve all the nodes of these families of plane graphs.

Download Full-text

Computation of Edge Resolvability of Benzenoid Tripod Structure

Journal of Mathematics ◽

10.1155/2021/9336540 ◽

2021 ◽

Vol 2021 ◽

pp. 1-8

Author(s):

Ali Ahmad ◽

Sadia Husain ◽

Muhammad Azeem ◽

Kashif Elahi ◽

M. K. Siddiqui

Keyword(s):

Fault Tolerant ◽

Pharmaceutical Research ◽

Chemical Compounds ◽

Metric Dimension ◽

Resolving Set ◽

Unique Representation ◽

Chemical Structures ◽

Distance Vector

In chemistry, graphs are commonly used to show the structure of chemical compounds, with nodes and edges representing the atom and bond types, respectively. Edge resolving set λ e is an ordered subset of nodes of a graph C , in which each edge of C is distinctively determined by its distance vector to the nodes in λ . The cardinality of a minimum edge resolving set is called the edge metric dimension of C . An edge resolving set L e , f of C is fault-tolerant if λ e , f ∖ b is also an edge resolving set, for every b in λ e , f . Resolving set allows obtaining a unique representation for chemical structures. In particular, they were used in pharmaceutical research for discovering patterns common to a variety of drugs. In this paper, we determine the exact edge metric and fault-tolerant edge metric dimension of benzenoid tripod structure and proved that both parameters are constant.

Download Full-text

On the fault-tolerant metric dimension of certain interconnection networks

Journal of Applied Mathematics and Computing ◽

10.1007/s12190-018-01225-y ◽

2018 ◽

Vol 60 (1-2) ◽

pp. 517-535 ◽

Cited By ~ 6

Author(s):

Hassan Raza ◽

Sakander Hayat ◽

Xiang-Feng Pan

Keyword(s):

Interconnection Networks ◽

Fault Tolerant ◽

Metric Dimension

Download Full-text

Fault-Tolerant Resolvability and Extremal Structures of Graphs

Mathematics ◽

10.3390/math7010078 ◽

2019 ◽

Vol 7 (1) ◽

pp. 78 ◽

Cited By ~ 7

Author(s):

Hassan Raza ◽

Sakander Hayat ◽

Muhammad Imran ◽

Xiang-Feng Pan

Keyword(s):

Fault Tolerant ◽

Upper Bounds ◽

Metric Dimension ◽

Regular Graphs ◽

Lower And Upper Bounds ◽

Constant Fault

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, n − 1 , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure.

Download Full-text

The Fault-Tolerant Metric Dimension of Cographs

Fundamentals of Computation Theory - Lecture Notes in Computer Science ◽

10.1007/978-3-030-25027-0_24 ◽

2019 ◽

pp. 350-364

Author(s):

Duygu Vietz ◽

Egon Wanke

Keyword(s):

Fault Tolerant ◽

Metric Dimension

Download Full-text

Fault-tolerant metric dimension problem: A new integer linear programming formulation and exact formula for grid graphs

Kragujevac Journal of Mathematics ◽

10.5937/kgjmath1804495s ◽

2018 ◽

Vol 42 (4) ◽

pp. 495-503 ◽

Cited By ~ 1

Author(s):

Ana Simić ◽

Milena Bogdanović ◽

Zoran Maksimović ◽

Jelisavka Milošević

Keyword(s):

Linear Programming ◽

Integer Linear Programming ◽

Fault Tolerant ◽

Exact Formula ◽

Metric Dimension ◽

Programming Formulation ◽

Grid Graphs ◽

Linear Programming Formulation ◽

Integer Linear Programming Formulation

Download Full-text

Fault-Tolerant Metric Dimension of Circulant Graphs

Mathematics ◽

10.3390/math10010124 ◽

2022 ◽

Vol 10 (1) ◽

pp. 124

Author(s):

Laxman Saha ◽

Rupen Lama ◽

Kalishankar Tiwary ◽

Kinkar Chandra Das ◽

Yilun Shang

Keyword(s):

Connected Graph ◽

Fault Tolerant ◽

Metric Dimension ◽

Resolving Set ◽

Circulant Graphs ◽

Minimum Cardinality ◽

Vertex Set

Let G be a connected graph with vertex set V(G) and d(u,v) be the distance between the vertices u and v. A set of vertices S={s1,s2,…,sk}⊂V(G) is called a resolving set for G if, for any two distinct vertices u,v∈V(G), there is a vertex si∈S such that d(u,si)≠d(v,si). A resolving set S for G is fault-tolerant if S\{x} is also a resolving set, for each x in S, and the fault-tolerant metric dimension of G, denoted by β′(G), is the minimum cardinality of such a set. The paper of Basak et al. on fault-tolerant metric dimension of circulant graphs Cn(1,2,3) has determined the exact value of β′(Cn(1,2,3)). In this article, we extend the results of Basak et al. to the graph Cn(1,2,3,4) and obtain the exact value of β′(Cn(1,2,3,4)) for all n≥22.

Download Full-text