Metric Dimension Of Graph Research Articles

On metric dimension of some planar graphs with 2n odd sided faces

Let [Formula: see text] be a connected graph of size [Formula: see text]. If [Formula: see text] is an ordered subset of distinct vertices in [Formula: see text] then the subset [Formula: see text] is said to be a resolving set for [Formula: see text], if all of the vertices in the graph can be uniquely defined by the vector of distances to the vertices in [Formula: see text]. A resolving set [Formula: see text] with minimum possible vertices is said to be a metric basis for [Formula: see text]. The cardinality of the metric basis is called the metric dimension of the graph [Formula: see text]. In this paper, we demonstrate that the metric dimension for some families of related planar graphs is three.

Binary Equilibrium Optimization Algorithm for Computing Connected Domination Metric Dimension Problem

We consider, in this paper, the NP-hard problem of finding the minimum connected domination metric dimension of graphs. A vertex set B of a connected graph G = (V, E) resolves G if every vertex of G is uniquely identified by its vector of distances to the vertices in B. A resolving set B of G is connected if the subgraphB¯induced by B is a nontrivial connected subgraph of G. A resolving set is dominating if every vertex of G that does not belong to B is a neighbor to some vertices in B. The cardinality of the smallest resolving set of G, the cardinality of the minimal connected resolving set, and the cardinality of the minimal connected domination resolving set are the metric dimension of G, connected metric dimension of G, and connected domination metric dimension of G, respectively. We present the first attempt to compute heuristically the minimum connected dominant resolving set of graphs by a binary version of the equilibrium optimization algorithm (BEOA). The particles of BEOA are binary-encoded and used to represent which one of the vertices of the graph belongs to the connected domination resolving set. The feasibility is enforced by repairing particles such that an additional vertex generated from vertices of G is added to B, and this repairing process is iterated until B becomes the connected domination resolving set. The proposed BEOA is tested using graph results that are computed theoretically and compared to competitive algorithms. Computational results and their analysis show that BEOA outperforms the binary Grey Wolf Optimizer (BGWO), the binary Particle Swarm Optimizer (BPSO), the binary Whale Optimizer (BWO), the binary Slime Mould Optimizer (BSMO), the binary Grasshopper Optimizer (BGO), the binary Artificial Ecosystem Optimizer (BAEO), and the binary Elephant Herding Optimizer (BEHO).

Truncated metric dimension for finite graphs

Let G be a graph with vertex set V(G), and let d(x,y) denote the length of a shortest path between nodes x and y in G. For a positive integer k and for distinct x,y∈V(G), let dk(x,y)=min{d(x,y),k+1} and Rk{x,y}={z∈V(G):dk(x,z)≠dk(y,z)}. A subset S⊆V(G) is a k-truncated resolving set of G if |S∩Rk{x,y}|≥1 for any pair of distinct x,y∈V(G). The k-truncated metric dimension, dimk(G), of G is the minimum cardinality over all k-truncated resolving sets of G, and the usual metric dimension is recovered when k+1 is at least the diameter of G. We obtain some general bounds for k-truncated metric dimension. For all k≥1, we characterize connected graphs G of order n with dimk(G)=n−2 and dimk(G)=n−1. For all j,k≥1, we find the maximum possible order, degree, clique number, and chromatic number of any graph G with dimk(G)=j. We determine dimk(G) when G is a cycle or a path. We also examine the effect of vertex or edge deletion on the truncated metric dimension of graphs, and study various problems related to the truncated metric dimension of trees.

Remarks on the Vertex and the Edge Metric Dimension of 2-Connected Graphs

The vertex (respectively edge) metric dimension of a graph G is the size of a smallest vertex set in G, which distinguishes all pairs of vertices (respectively edges) in G, and it is denoted by dim(G) (respectively edim(G)). The upper bounds dim(G)≤2c(G)−1 and edim(G)≤2c(G)−1, where c(G) denotes the cyclomatic number of G, were established to hold for cacti without leaves distinct from cycles, and moreover, all leafless cacti that attain the bounds were characterized. It was further conjectured that the same bounds hold for general connected graphs without leaves, and this conjecture was supported by showing that the problem reduces to 2-connected graphs. In this paper, we focus on Θ-graphs, as the most simple 2-connected graphs distinct from the cycle, and show that the the upper bound 2c(G)−1 holds for both metric dimensions of Θ-graphs; we characterize all Θ-graphs for which the bound is attained. We conclude by conjecturing that there are no other extremal graphs for the bound 2c(G)−1 in the class of leafless graphs besides already known extremal cacti and extremal Θ-graphs mentioned here.

Learning to compute the metric dimension of graphs

For distinct vertices x,y,z in graph G=(V,E), we say the pair of vertices x,y can be distinguished by vertex z, if the distance inequation dG(x,z)≠dG(y,z) is true. If any two different vertices in graph G can be distinguished by at least one vertex in the vertex subset S⊂V then we say S⊂V is a resolving set of graph G. The one with minimum number of nodes among all of the resolving sets is called a metric base of graph G, of which the cardinality is called the metric dimension of graph G. The metric dimension problem(MDP) of computing the metric dimension of graphs is a kind of complex combinatorial optimization problem. In this paper, a hybrid algorithm for solving the MDP is developed, through which the metric base of graphs can be learned, and consequently the metric dimension of graphs is estimated. It is particular that the hybrid algorithm combines graph representation learning with greedy repair policy. Extensive experiments show that the hybrid algorithm has higher computational accuracy and efficiency on random graphs. It is also further suggested that the learning process of MDP is greatly influenced by the attributes of complex networks and the designs of graph neural networks themselves.

On Adjacency Metric Dimension of Some Families of Graph

Metric dimension of a graph is a well-studied concept. Recently, adjacency metric dimension of graph has been introduced. A set Q a ⊂ V G is considered to be an adjacency metric generator for G if u 1 , u 2 ∈ V \ Q a (supposing each pair); there must exist a vertex q ∈ Q a along with the condition that q is indeed adjacent to one of u 1 , u 2 . The minimum number of elements in adjacency metric generator is the adjacency metric dimension of G , denoted by di m a G . In this work, we compute exact values of the adjacency metric dimension of circulant graph C n 1 , 2 , Möbius ladder, hexagonal Möbius ladder, and the ladder graph.

Metric Dimension of Graphs and its Application to Robotic Navigation

Metric Dimension of Graphs and its Application to Robotic Navigation

The local complement metric dimension of graphs

The metric dimension of graphs has been extended in some types and variations such as local metric dimension and complement metric dimension of graphs. Merging two concepts is one way of developing the concept of graph theory as a branch of mathematics. These two variations motivated us to construct a new concept of metric dimension so called local complement metric dimension. We apply this new concept to some particular classes of graphs and the corona product of two graphs. We also characterize the local complement metric dimension of graphs with certain properties, namely, bipartite graphs and odd cycle graphs. Furthermore, we discover the local complement metric dimension of the corona product of two particular graphs as well as the corona product of two general graphs.

Metric Dimension of Some Generalized Families of Toeplitz Graphs

The metric dimension of a graph G is the selection of the minimum possible number of vertices such that each vertex of the graph G is distinctively defined by its vector of distances to the set of selected vertices. It was proved that the problem of determining the metric dimension of a graph is NP-hard. In this paper, the metric dimension of Toeplitz graphs with two and three generators is discussed and the exact values are found. Also, two conjectures about the exact metric dimension of Toeplitz graphs are given.

The effect of vertex and edge deletion on the edge metric dimension of graphs

The effect of vertex and edge deletion on the edge metric dimension of graphs

The mixed metric dimension of wheel-like graphs

Consider the graph G = (V, E). It is a connected graph. It is a simple graph too. A node w ∈ V, then we call vertex, determined two elements of graph. There are vertices and edges of graphs. Any two vertices x, y ∈ E ∪ V if d(w, x) ≠ d(w, y), which d(w, x) and d(w, y) is the mixed distance of the element w (vertices or edges) in graph G. A set of vertices in a graph G is represented by the symbol Rm that defines a mixed metric generator for G, if the elements of vertices or edges are stipulated by several vertex set of Rm . There’s a chance that some mixed metric generators have varied cardinality. We choose one whose the minimum cardinality and it is called the mixed metric dimension of graph G, denoted by dimm (G). This research examines the mixed metric dimension of gear Gn , helm Hn , sunflower SFn , and friendship graph Frn . We call these graphs by wheel-like graphs. Our findings include the mixed metric dimension of gear graph Gn of order n ≥ 4 is dimm (Gn ) = n, helm graph Hn of order n ≥ 4 is dimm (Hn ) = n, sunflower graph SFn of order n ≥ 5 is dimm (SFn ) = n and friendship graph Frn of order n ≥ 3 is dimm (Frn ) = 2n.

Metric Dimension Threshold of Graphs

Let G be a connected graph. A subset S of vertices of G is said to be a resolving set of G, if for any two vertices u and v of G there is at least a member w of S such that d(u, w) ≠ d(v, w). The minimum number t that any subset S of vertices G with |S| = t is a resolving set for G, is called the metric dimension threshold, and is denoted by dimth(G). In this paper, the concept of metric dimension threshold is introduced according to its application in some real‐word problems. Also, the metric dimension threshold of some families of graphs and a characterization of graphs G of order n for which the metric dimension threshold equals 2, n − 2, and n − 1 are given. Moreover, some graphs with equal the metric dimension threshold and the standard metric dimension of graphs are presented.

Local metric dimension for graphs with small clique numbers

Let G be a connected graph. Given a set S={v1,…,vk}⊆V(G), the metric S-representation of a vertex v in G is the vector (dG(v,v1),…,dG(v,vk)) where dG(a,b) is the distance between vertices a and b in G. If every two adjacent vertices of G have different metric S-representations, then the set S is a local metric generator for G. The local metric dimension of G, denoted by dimℓ⁡(G), is the minimum cardinality among all local metric generators of G. The local metric dimension of graphs with large clique numbers is known. Our contribution in this paper is to present upper bounds on the local metric dimension of connected graphs with small clique numbers. We show that there exists a constant C such that if G is a fullerene graph of order n, then dimℓ⁡(G)≤min⁡{C,125n+1}. As a consequence of this result, we show that determining the local metric dimension of fullerene graphs can be done in polynomial time. If G is a connected triangle-free graph of order n, then we show that dimℓ⁡(G)≤γ(G), where γ(G) is the domination number of G. As an application of this result, we prove that dimℓ⁡(G)≤25n. If G is a connected graph of order n with clique number 3, then we show that dimℓ⁡(G)≤n−α′(G), where α′(G) is the matching number of G. As an application of this result, we show that for such graphs G we have dimℓ⁡(G)≤23n.

Some binary products and integer linear programming for [formula omitted]-metric dimension of graphs

A sharp upper bound and a closed formula for the k-metric dimension of the hierarchical product of graphs is proved. Also, sharp lower bounds for the k-metric dimension of the splice and link products of graphs are presented. An integer linear programming model for computing the k-metric dimension as well as a k-metric basis of a given graph is proposed. These results are applied to bound or to compute the k-metric dimension of some classes of graphs that are of interest in mathematical chemistry.

Graphs of Neighborhood Metric Dimension Two

A subset of vertices of a simple connected graph is a neighborhood set (n-set) of G if G is the union of subgraphs of G induced by the closed neighbors of elements in S. Further, a set S is a resolving set of G if for each pair of distinct vertices x,y of G, there is a vertex s∈ S such that d(s,x)≠d(s,y). An n-set that serves as a resolving set for G is called an nr-set of G. The nr-set with least cardinality is called an nr-metric basis of G and its cardinality is called the neighborhood metric dimension of graph G. In this paper, we characterize graphs of neighborhood metric dimension two.

Mixed metric dimension of graphs with edge disjoint cycles

In a connected graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V(G)∪E(G) is called the mixed metric dimension of G. In this paper we first establish the exact value of the mixed metric dimension of a unicyclic graph G which is derived from the structure of G. We further consider graphs G with edge disjoint cycles, where for each cycle Ci of G we define a unicyclic subgraph Gi of G in which Ci is the only cycle. Applying the result for unicyclic graph to the subgraph Gi of every cycle Ci then yields the exact value of the mixed metric dimension of such a graph G. The obtained formulas for the exact value of the mixed metric dimension yield a simple sharp upper bound on the mixed metric dimension, and we conclude the paper conjecturing that the analogous bound holds for general graphs with prescribed cyclomatic number.

Dominant Local Metric Dimension of Wheel Related Graphs

Dominant local metric dimension is consist of two interesting topic in graph theory, they are dominating and metric dimension which be expanded as local metric dimension. A graph G is said having dominant local metric dimension if G be a connected graph and there is an ordered subset W l = {W 1, W 2 …, W n }⊆V(G) where W l is a local resolving set as well as a dominating set of G. Minimum cardinality of dominant local resolving set of G is called the dominant local basis of G. Then, cardinality of the dominant local basis of G is called the dominant local metric dimension of G which denoted by Ddim l (G). In this paper, we determine the dominant local metric dimension of wheel related graphs. That are Wheel, Jahangir, Friendship and Helm graph. Furthermore, we characterize the dominant local metric dimension of those graphs.

On the resolving strong domination number of graphs: a new notion

The study of metric dimension of graph G has widely given some results and contribution of graph research of interest, including the domination set theory. The dominating set theory has been quickly growing and there are a lot of natural extension of this study, such as vertex domination, edge domination, total domination, power domination as well as the strong domination. In this study, we initiate to combine the two above concepts, namely metric dimension and strong domination set. Thus we have a resolving strong domination set. We have obtained the resolving strong domination number, denoted by γrst(G), of some graphs.

Graphs with the edge metric dimension smaller than the metric dimension

Given a connected graph G, the metric (resp. edge metric) dimension of G is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of G by means of distance vectors to such a set. In this work, we settle three open problems on (edge) metric dimension of graphs. Specifically, we show that for every r,t≥2 with r≠t, there is n0, such that for every n≥n0 there exists a graph G of order n with metric dimension r and edge metric dimension t, which among other consequences, shows the existence of infinitely many graph whose edge metric dimension is strictly smaller than its metric dimension. In addition, we also prove that it is not possible to bound the edge metric dimension of a graph G by some constant factor of the metric dimension of G.

On the Dominant Local Metric Dimension of Graphs

Mathematics is used throughout the world as an essential tool in many fields. One of the new concepts in Graph Theory as the branch of mathematics is defined in this paper which is called the dominant local metric dimension. Let G be a connected graph. The ordered subsetWl = {w1, w2, w3, …, wn} ⊆ V(G) is called a dominant local resolving set of G if Wl is a local resolving set as well as a dominating set of G. A dominant local resolving set of G with minimum cardinality is called the dominant local basis of G. The cardinality of the dominant local basis of G is called the dominant local metric dimension of G and denoted by Ddiml(G). In this article, we present the methods for determining the dominant local metric dimension of graphs, characterize the dominant local metric dimension of graph G, and also determine the dominant local metric dimension of some particular classes of graphs.