On the Families of Graphs with Unbounded Metric Dimension (original) (raw)

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The simultaneous metric dimension of graph families

Discrete Applied Mathematics, 2015

A vertex v ∈ V is said to resolve two vertices x and y if dG(v, x) = dG (v, y). A set S ⊂ V is said to be a metric generator for G if any pair of vertices of G is resolved by some element of S. A minimum metric generator is called a metric basis, and its cardinality, dim(G), the metric dimension of G. A set S ⊆ V is said to be a simultaneous metric generator for a graph family G = {G1, G2, . . . , G k }, defined on a common (labeled) vertex set, if it is a metric generator for every graph of the family. A minimum cardinality simultaneous metric generator is called a simultaneous metric basis, and its cardinality the simultaneous metric dimension of G. We obtain sharp bounds for this invariants for general families of graphs and calculate closed formulae or tight bounds for the simultaneous metric dimension of several specific graph families. For a given graph G we describe a process for obtaining a lower bound on the maximum number of graphs in a family containing G that has simultaneous metric dimension equal to dim(G). It is shown that the problem of finding the simultaneous metric dimension of families of trees is N P -hard. Sharp upper bounds for the simultaneous metric dimension of trees are established. The problem of finding this invariant for families of trees that can be obtained from an initial tree by a sequence of successive edge-exchanges is considered. For such families of trees sharp upper and lower bounds for the simultaneous metric dimension are established.

On the metric dimension of some families of graphs

Electronic Notes in Discrete Mathematics, 2005

The concept of (minimum) resolving set has proved to be useful and/or related to a variety of fields such as Chemistry [3,6], Robotic Navigation and Combinatorial Search and Optimization . This work is devoted to evaluating the so-called metric dimension of a finite connected graph, i.e., the minimum cardinality of a resolving set, for a number of graph families, as long as to study its behavior with respect to the join and the cartesian product of graphs.

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.

On the metric dimension of infinite graphs

Discrete Applied Mathematics, 2012

A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we study the metric dimension of infinite graphs such that all its vertices have finite degree. We give necessary conditions for those graphs to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some results about the metric dimension of the cartesian product of finite and infinite graphs, and give the metric dimension of the cartesian product of several families of graphs.

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.

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.

On the metric dimension of Cartesian product of graphs

2007

A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G H. We prove that the metric dimension of G G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G G is unbounded. 2000 Mathematics Subject Classification. 05C12 (distance in graphs).

Metric dimension of growing infinite graphs

2022

We investigate how the metric dimension of infinite graphs change when we add edges to the graph. Our two main results: (1) there exists a growing sequence of graphs (under the subgraph relation, but without adding vertices) for which the metric dimension changes between finite and infinite infinitely many times; (2) finite changes in the edge set can not change the metric dimension from finite to infinite or vice versa.

On the Metric Dimension of Cartesian Products of Graphs

Siam Journal on Discrete Mathematics, 2007

A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G H. We prove that the metric dimension of G G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G G is unbounded.