On Cycle Related Graphs with Constant Metric Dimension (original) (raw)
The connected metric dimension at a vertex of a graph
Theoretical Computer Science, 2020
The notion of metric dimension, dim(G), of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing cdim G (v), the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex z ∈ S such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, cdim G (v) is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by cdim(G), is min{cdim G (v) : v ∈ V (G)}. Noting that 1 ≤ dim(G) ≤ cdim(G) ≤ cdim G (v) ≤ |V (G)|−1 for any vertex v of G, we show the existence of a pair (G, v) such that cdim G (v) takes all positive integer values from dim(G) to |V (G)| − 1, as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying cdim G (v) ∈ {1, |V (G)| − 1}. We show that cdim(G) = 2 implies G is planar, whereas it is well known that there is a non-planar graph H with dim(H) = 2. We also characterize trees and unicyclic graphs G satisfying cdim(G) = dim(G). We show that cdim(G) − dim(G) can be arbitrarily large. We determine cdim(G) and cdim G (v) for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems.
Cycle-balance conditions for distance-regular graphs
Discrete Mathematics, 2003
In a distance-regular graph, the partition with respect to distance from a vertex supports a unique eigenvector for each eigenvalue. There may be non-singleton vertex sets whose corresponding distance partition also supports eigenvectors. We consider the members of three families of distance regular graphs, the Johnson Graphs, Hamming Graphs and Complete Multipartite graphs. For each we determine all such sets which support an eigenvector for the next to largest eigenvalue. These sets exhibit the underlying geometric structure of the graph.
On Metric Dimension of Some Rotationally Symmetric Graphs
ISRN Combinatorics, 2013
A family 𝒢 of connected graphs is 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 the graph An∗ and the graph Anp obtained from the antiprism graph have constant metric dimension.
Studies in graph theory distance related concepts in graphs
2013
By a graph G = (V,E), we mean a finite undirected graph with neither loops nor multiple edges. The order and size of G are denoted by n = |V | and m = |E| respectively. For graph theoretic terminology we refer to Chartrand and Lesniak [7]. In Chapter 1, we collect some basic definitions and theorems on graphs which are needed for the subsequent chapters. The distance d(u, v) between two vertices u and v of a connected graph G is the length of a shortest u-v path in G. There are several distance related concepts and parameters such as eccentricity, radius, diameter, convexity and metric dimension which have been investigated by several authors in terms of theory and applications. An excellent treatment of various distances and distance related parameters are given in Buckley and Harary [6]. Let G = (V,E) be a graph. Let v ∈ V . The open neighborhood N(v) of a vertex v is the set of vertices adjacent to v. Thus N(v) = {w ∈ V : wv ∈ E}. The closed neighborhood of a vertex v, is the set...
Metric Dimensions of Bicyclic Graphs
Mathematics, 2023
The distance d(v a , v b ) between two vertices of a simple connected graph G is the length of the shortest path between v a and v b . Vertices v a , v b of G are considered to be resolved by a vertex v if The representation of vertex v with respect to W is denoted by r(v|W) and is an s-vector(s-tuple A minimal resolving set is termed a metric basis for G. The cardinality of the metric basis set is called the metric dimension of G, represented by dim(G). In this article, we study the metric dimension of two types of bicyclic graphs. The obtained results prove that they have constant metric dimension.
GRAPHS WITH METRIC DIMENSION TWO
International Journal of Advanced Research in Mathematics and Applications, 2014
In this paper, we discuss some characteristics of a graph due to the properties of distance partition and establish some results pertaining to the structure of a graph G with 2. Finally the graphs with 2 is characterized which is in fact proved in [15].
Edge metric dimension of some classes of circulant graphs
Analele Universitatii "Ovidius" Constanta - Seria Matematica
Let G = (V (G), E(G)) be a connected graph and x, y ∈ V (G), d(x, y) = min{ length of x − y path } and for e ∈ E(G), d(x, e) = min{d(x, a), d(x, b)}, where e = ab. A vertex x distinguishes two edges e 1 and e 2, if d(e 1, x) ≠ d(e 2, x). Let WE = {w 1, w 2, . . ., wk } be an ordered set in V (G) and let e ∈ E(G). The representation r(e | WE ) of e with respect to WE is the k-tuple (d(e, w 1), d(e, w 2), . . ., d(e, wk )). If distinct edges of G have distinct representation with respect to WE , then WE is called an edge metric generator for G. An edge metric generator of minimum cardinality is an edge metric basis for G, and its cardinality is called edge metric dimension of G, denoted by edim(G). The circulant graph Cn (1, m) has vertex set {v 1, v 2, . . ., vn } and edge set {vivi +1 : 1 ≤ i ≤ n−1}∪{vnv 1}∪{vivi+m : 1 ≤ i ≤ n−m}∪{vn−m+ivi : 1 ≤ i ≤ m}. In this paper, it is shown that the edge metric dimension of circulant graphs Cn (1, 2) and Cn (1, 3) is constant.
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.
Structural Analysis of Complex Networks, 2010
The distance between two vertices is the basis of the definition of several graph parameters including diameter, radius, average distance and metric dimension. These invariants are examined, especially how they relate to one another and to other graph invariants and their behaviour in certain graph classes. We also discuss characterizations of graph classes described in terms of distance or shortest paths. Finally, generalizations are considered.
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.