On Metric Dimension of Two Constructed Families from Antiprism Graph (original) (raw)

The Simultaneous Strong Metric Dimension of Graph Families

Bulletin of the Malaysian Mathematical Sciences Society, 2015

Let G be a family of graphs defined on a common (labeled) vertex set V. A set S ⊂ V is said to be a simultaneous strong metric generator for G if it is a strong metric generator for every graph of the family. The minimum cardinality among all simultaneous strong metric generators for G, denoted by Sd s (G), is called the simultaneous strong metric dimension of G. We obtain general results on Sd s (G) for arbitrary families of graphs, with special emphasis on the case of families composed by a graph and its complement. In particular, it is shown that the problem of finding the simultaneous strong metric dimension of families of graphs is N P-hard, even when restricted to families of trees.

The k-metric dimension of graphs: a general approach

arXiv (Cornell University), 2016

Let (X, d) be a metric space. A set S ⊆ X is said to be a k-metric generator for X if and only if for any pair of different points u, v ∈ X, there exist at least k points w 1 , w 2 ,. .. w k ∈ S such that d(u, w i) = d(v, w i), for all i ∈ {1,. .. k}. Let R k (X) be the set of metric generators for X. The k-metric dimension dim k (X) of (X, d) is defined as dim k (X) = inf{|S| : S ∈ R k (X)}. Here, we discuss the k-metric dimension of (V, d t), where V is the set of vertices of a simple graph G and the metric d t : V × V → N ∪ {0} is defined by d t (x, y) = min{d(x, y), t} from the geodesic distance d in G and a positive integer t. The case t ≥ D(G), where D(G) denotes the diameter of G, corresponds to the original theory of k-metric dimension and the case t = 2 corresponds to the theory of k-adjacency dimension. Furthermore, this approach allows us to extend the theory of k-metric dimension to the general case of non-necessarily connected graphs.

On the Metric Dimension of Imprimitive Distance-Regular Graphs

Annals of Combinatorics, 2016

A resolving set for a graph Γ is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of Γ is the smallest size of a resolving set for Γ. Much attention has been paid to the metric dimension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimension of primitive distanceregular graphs in terms of their parameters. We show how the metric dimension of an imprimitive distance-regular graph can be related to that of its halved and folded graphs, but also consider infinite families (including Taylor graphs and the incidence graphs of certain symmetric designs) where more precise results are possible.

The Simultaneous Metric Dimension of Families Composed by Lexicographic Product Graphs

Graphs and Combinatorics, 2016

Let G be a graph family defined on a common (labeled) vertex set V. A set S ⊆ V is said to be a simultaneous metric generator for G if for every G ∈ G and every pair of different vertices u, v ∈ V there exists s ∈ S such that dG(s, u) = dG(s, v), where dG denotes the geodesic distance. A simultaneous adjacency generator for G is a simultaneous metric generator under the metric dG,2(x, y) = min{dG(x, y), 2}. A minimum cardinality simultaneous metric (adjacency) generator for G is a simultaneous metric (adjacency) basis, and its cardinality the simultaneous metric (adjacency) dimension of G. Based on the simultaneous adjacency dimension, we study the simultaneous metric dimension of families composed by lexicographic product graphs.

THE COMPLEMENT METRIC DIMENSION OF GRAPHS AND ITS OPERATIONS

IAEME, 2019

Let G be a connected graph with vertex set V(G) and edge set E(G). The distance between vertices u and v in G is denoted by d(u, v), which serves as the shortest path length from u to v. Let 𝑊 = {𝑤1, 𝑤2, … , 𝑤𝑘 } ⊆ 𝑉(𝐺) be an ordered set, and v is a vertex in G. The representation of v with respect to W is an ordered set 𝑘 − 𝑡𝑢𝑝𝑙𝑒, 𝑟(𝑣|𝑊) = (𝑑(𝑣, 𝑤1), 𝑑(𝑣, 𝑤2), … , 𝑑(𝑣, 𝑤𝑘 )). The set W is called a resolving set for G if each vertex in G has a different representation with respect to W. A resolving set containing minimum cardinality is called a basis for G. The number of vertices in a basis of G is called metric dimension of G, which is denoted by𝑑𝑖𝑚(𝐺). The 𝑆 ⊆ 𝑉(𝐺) is a complement resolving set of G if there are two vertices𝑢, 𝑣 ∈ 𝑉(𝐺) ∖ 𝑆, such that𝑟(𝑢|𝑆) = 𝑟(𝑣|𝑆). A complement basis of G is the complement resolving set containing maximum cardinality. The number of vertices in a complement basis of G is called complement metric dimension of G, which is denoted by 𝑑̅̅𝑖̅𝑚̅̅(𝐺). In this paper, we examined complement metric dimension of particular graphs and their characteristics. Furthermore, we determined complement metric dimension of corona and comb products graphs

Families of Regular Graphs With Constant Metric Dimension

Utilitas Mathematica, 2008

Let G be a connected graph and d(x, y) be the distance between the vertices x and y. A subset of vertices W = {w1,. .. , w k } is called a resolving set for G if for every two distinct vertices x, y ∈ V (G), there is a vertex wi ∈ W such that d(x, wi) = d(y, wi). A resolving set containing a minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension dim(G). A family of connected graphs G is said to be a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in G. In this paper, we show that generalized Petersen graphs P (n, 2), antiprisms A n and Harary graphs H 4,n for n ≡ 1(mod 4) are families of regular graphs with constant metric dimension and raise some questions in a more general setting.

Bi-Edge Metric Dimension of Graphs

IJCSAM (International Journal of Computing Science and Applied Mathematics), 2024

Given a connected G = (V (G), E(G)) graph. The main problem in graph metric dimensions is calculating the metric dimensions and their characterization. In this research, a new dimension concept is introduced, namely a bi-edge metric dimension of graph which is a development of the concpet of bi-metric graphs with the innovation of bi-metric graph representations to become the bi-edge metric graph representations. In this case, what is meant by bi-edge metric and edge detour. If there is a set in G that causes every edge in G has a different bi-edge metric representation in G, then that set is called the biedge metric resolving set. The minimum cardinality of the bi-edge metric resolving set graphs is called the bi-edge metric dimension of G graph, denoted by e dim b (G). The spesific purpose of this research is to apply the concept of bi-edge metric dimensions to special graphs, such as cycle, complete, star and path can be obtained.

The Simultaneous Local Metric Dimension of Graph Families

Symmetry

In a graph G = (V, E), a vertex v ∈ V is said to distinguish two vertices x and y if d G (v, x) = d G (v, y). A set S ⊆ V is said to be a local metric generator for G if any pair of adjacent vertices of G is distinguished by some element of S. A minimum local metric generator is called a local metric basis and its cardinality the local metric dimension of G. A set S ⊆ V is said to be a simultaneous local metric generator for a graph family G = {G 1 , G 2 ,. .. , G k }, defined on a common vertex set, if it is a local metric generator for every graph of the family. A minimum simultaneous local metric generator is called a simultaneous local metric basis and its cardinality the simultaneous local metric dimension of G. We study the properties of simultaneous local metric generators and bases, obtain closed formulae or tight bounds for the simultaneous local metric dimension of several graph families and analyze the complexity of computing this parameter.

The metric dimensions of bridge graphs for some classes of graphs

2020

There are many open problems in the metric dimension of a graph, espessially the bridge graph and the disconnected graph, that have not been resolved until now. This paper presents the metrics dimension of the bridge graph in several classes of graphs namely cycle, complete, and star graphs. We know that the metric dimensions of the complete, cycle or star graph have been obtained. The bridge graph B (G1, G2, e) is a graph which is obtained from the operation of adding edge e to graphs G1 and G2. To obtain the metric dimension of the bridge graph from the graphs G1 and G2 on edge e, pd(B (G1, G2, e)), we used the graph structure of G1, G2 and the properties of the bridge graph based on the endpoint on the edge e. The results obtained the metric dimension of the bridge graph B(G1, G2, e) for the cycle, complete, or star graphs.

The Edge Version of Metric Dimension for the Family of Circulant Graphs Cₙ(1, 2)

IEEE Access, 2021

Graph theory is widely used to analyze the structure models in chemistry, biology, computer science, operations research and sociology. Molecular bonds, species movement between regions, development of computer algorithms, shortest spanning trees in weighted graphs, aircraft scheduling and exploration of diffusion mechanisms are some of these structure models. Let G = (V G , E G) be a connected graph, where V G and E G represent the set of vertices and the set of edges respectively. The idea of the edge version of metric dimension is based on the distance of edges in a graph. Let R E G be the smallest set of edges in a connected graph G that forms a basis such that for every pair of edges e 1 , e 2 ∈ E G , there exists an edge e ∈ R E G for which d E G (e 1 , e) = d E G (e 2 , e) holds. In this paper, we show that the family of circulant graphs C n (1, 2) is the family of graphs with constant edge version of metric dimension. INDEX TERMS Line graph, resolving sets, the edge version of metric dimension, circulant graphs.