On the central resolver set of the edge coronation graphs

Let G be a simple connected graph. A metric dimension s of a graph G is the cardinality of the smallest subset S of vertices such that all other vertices are uniquely determined by their distances to the vertices in S. A vertex w of graph G strongly resolves two vertices u, v ∈ V (G) if one of the equalities hold: dG(w, u) = dG(w, v) + dG(v, u) or dG(w, v) = dG(w, u) + dG(u, v). In other words, a vertex w in a graph G strongly resolves a pair of vertices u, v if there exists a shortest w—u path containing v or a shortest w—v path containing u in G. A set S of minimum cardinality whose elements strongly resolve any pair of vertices of G is called a strong metric basis of graph G. Typical, a metric dimension of a graph G is not equal to a strong metric dimension of a graph G. A metric dimension as a graph parameter and strong metric dimension have numerous applications. In general, a search of metric dimension and strong metric dimension is NP-hard problem. But for some families of graphs, for example, for trees, there is a polynomial algorithm for that searching. This paper characterizes such trees that a metric dimension equals a strong metric dimension.In this article, we use properties of strong metric basis of trees and cycles to obtain a closed formula for calculating a strong metric dimension of unicyclic graphs, namely graphs that have only one cycle. We say that leaf u which lies out of the cycle is projected onto vertex v that lies in the cycle if deg(v) ≥ 3 and v is connected to u through the shortest path. A strong metric dimension depends on the number of inner vertexes of the cycle, their position in it, and the number of leaves that are projected onto each inner vertex of the cycle. Note that now there is no closed formula for calculating metric dimension of unicyclic graphs.

The strong local metric dimension is the development result of a strong metric dimension study, one of the study topics in graph theory. Some of graphs that have been discovered about strong local metric dimension are path graph, star graph, complete graph, cycle graphs, and the result corona product graph. In the previous study have been built about strong local metric dimensions of corona product graph. The purpose of this research is to determine the strong local metric dimension of cartesian product graph between any connected graph G and H, denoted by dimsl (G x H). In this research, local metric dimension of G x H is influenced by local strong metric dimension of graph G and local strong metric dimension of graph H. Graph G and graph H has at least two order.

In this paper, we study the fault-tolerant metric dimension in graph theory, an important measure against failures in unique vertex identification. The metric dimension of a graph is the smallest number of vertices required to uniquely identify every other vertex based on their distances from these chosen vertices. Building on existing work, we explore fault tolerance by considering the minimal number of vertices needed to ensure that all other vertices remain uniquely identifiable even if a specified number of these vertices fails. We compute the fault-tolerant metric dimension of various chemical graphs, namely fullerenes, benzene, and polyphenyl graphs.

In this paper, we investigate bounds for the fault-tolerant metric dimension and adjacency fault-tolerant resolving set of corona product graphs. Let [Formula: see text] and [Formula: see text] be two graphs with orders [Formula: see text] and [Formula: see text], respectively. The corona product of the graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is constructed by taking one copy of [Formula: see text] and [Formula: see text] copies of [Formula: see text], connecting each vertex of the [Formula: see text]th copy of [Formula: see text] to the [Formula: see text]th vertex of [Formula: see text] by an edge. For any integer [Formula: see text], we recursively define the graph [Formula: see text] as [Formula: see text]. We present several results on the fault-tolerant metric dimension and the adjacency fault-tolerant resolving set of corona product graphs. Additionally, we explore the relationship between the fault-tolerant metric dimension and the adjacency fault-tolerant metric dimension of a graph [Formula: see text]. Finally, we determine the adjacency fault-tolerant metric dimension of path, cycle, complete, complete bipartite, and star graphs.

Basically, the new topic of the dominant local metric dimension which be symbolized by Ddim_l (H) is a combination of two concepts in graph theory, they were called the local metric dimension and dominating set. There are some terms in this topic that is dominant local resolving set and dominant local basis. An ordered subset W_l is said a dominant local resolving set of G if W_l is dominating set and also local resolving set of G. While dominant local basis is a dominant local resolving set with minimum cardinality. This study uses literature study method by observing the local metric dimension and dominating number before detecting the dominant local metric dimension of the graphs. After obtaining some new results, the purpose of this research is how the dominant local metric dimension of vertex amalgamation product graphs. Some special graphs that be used are star, friendship, complete graph and complete bipartite graph. Based on all observation results, it can be said that the dominant local metric dimension for any vertex amalgamation product graph depends on the dominant local metric dimension of the copied graphs and how the terminal vertex is constructed

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.

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity The metric dimension of a graph Gamma is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph G(q)(n, k) (whose vertices are the k-subspaces of F-q(n), and are adjacent if they intersect in a (k 1)-subspace) for k \textgreater= 2. We find an upper bound on its metric dimension, which is equal to the number of 1-dimensional subspaces of F-q(n). We also give a construction of a resolving set of this size in the case where k + 1 divides n, and a related construction in other cases.

A set of vertices W is a resolving set of a graph G if every two vertices of G have distinct representations of distances with respect to the set W. The number of vertices in a smallest resolving set is called the metric dimension. This invariant has extensive applications in robotics, since the metric dimension can represent the mínimum number of landmarks, which uniquely determine the position of a robot moving in a graph space. Finding the metric dimension of a graph is an NP-hard problem. We present exact values of the metric dimensión of Kayak Paddles graph and Cycles with chord.

Learning to compute the metric dimension of graphs

The complement metric dimension of graph is one of the recent topics in graph theory. The concept came from the metric dimension which is a topic that has developed very rapidly. The complement metric dimension of graph G is denoted by dim¯(G). The goal of this research is to determine complement metric dimension of comb product of special graphs, such as path graph (Pn), star graph (Sn), and complete graph (Kn). Furthermore, we find complement metric dimension of comb product of any graphs G and H. We get that complement metric dimension of comb product of graph G and H depends on the order of both graph and complement metric dimension of graph H.

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.

The central local metric dimension is a new variation of local metric dimension that introduced in 2023. The central local metric dimension is a new concept that enriches research studies in graph theory, especially in the field of metric dimension. This concept combines the concept of local metric dimension by involving central points in the local metric set so that the existence of central points can strengthen the position of the local metric set in distinguishing every two neighboring points in a graph. The methodology of this research is study literature and observation. We find the central vertex of each graph and also find the local metric set of its graph, then we applied it to the related theorem to find the lower bound of central local metric dimension. Let 𝐺 be a connected graph with order 𝑛 and vertex set is 𝑉(𝐺). A subset 𝑊={𝑥1,𝑥2,…,𝑥𝑘}⊆𝑉(𝐺) is a local metric set of graph 𝐺 if the metric code of every two adjacent vertices 𝑢,𝑣 in 𝐺 are 𝑟(𝑢|𝑊)≠𝑟(𝑣|𝑊), where 𝑟(𝑢|𝑊)=(𝑑(𝑢,𝑥1),𝑑(𝑢,𝑥2),…,𝑑(𝑢,𝑥𝑘)) and 𝑟(𝑣|𝑊)=(𝑑(𝑣,𝑥1),𝑑(𝑣,𝑥2),…,𝑑(𝑣,𝑥𝑘)). A vertex 𝑥∈𝑉(𝐺) is a central vertex in 𝐺 if 𝑥 have the the shorthest distance to the another all vertices in 𝐺. If 𝑊 consist of all central vertices in 𝐺, then 𝑊 is called a central local metric set of 𝐺. The minimal cardinality of 𝑊 is called a central basis local set of 𝐺 and its cardinality is called central local metric dimension of 𝐺 or denoted by 𝑙𝑚𝑑𝑠(𝐺). In this paper we explored the central local metric dimension on generalized fan graph, generalized broken fan graph, and a graph resulting from corona operation Cm ⊙ K¯n. The result show that the central local metric dimension of generalized fan graph is equal with its order because the diameter and radius are equal. Different with it, the central local metric dimension of generalized broken fan is equal with its local metric dimension plus cardinality of central set, and the central local metric dimension of Cm ⊙ K¯n are equal with the central local metric dimension of 𝐶𝑚.

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the metric dimension of the random graph $G(n,p)$ for a wide range of probabilities $p=p(n)$.

Let G = (V, E) be a simple and connected graph. For each x ∈ V(G), it is associated with a vector pair (a, b), denoted by S x , corresponding to subset S = {s1 , s2 , ... , s k } ⊆ V(G), with a = (d(x, s1 ), d(x, s2 ), ... , d(x, s k )) and b = (δ(x, s1 ), δ(x, s2 ), ... , δ(x, s k )). d(v, s) is the length of shortest path from vertex v to s, and δ(v, s) is the length of the furthest path from vertex v to s. The set S is called the bi-resolving set in G if S x ≠ S y for any two distinct vertices x, y ∈ V(G). The bi- metric dimension of graph G, denoted by β b (G), is the minimum cardinality of the bi-resolving set in graph G. In this study we analyze bi-metric dimension in the antiprism graph (A n ). From the analysis that has been done, it is obtained the result that bi-metric dimension of graph A n , β b (A n ) is 3. Keywords: Antiprism graph, bi-metric dimension, bi-resolving set. .

The fractional versions of graph-theoretic invariants multiply the range of applications in scheduling, assignment and operational research problems. For this interesting aspect of fractional graph theory, we introduce the fractional version of local metric dimension of graphs. The local resolving neighborhood $L(xy)$ of an edge $xy$ of a graph $G$ is the set of those vertices in $G$ which resolve the vertices $x$ and $y$. A function $f:V(G)\rightarrow[0, 1]$ is a local resolving function of $G$ if $f(L(xy))\geq1$ for all edges $xy$ in $G$. The minimum value of $f(V(G))$ among all local resolving functions $f$ of $G$ is the fractional local metric dimension of $G$. We study the properties and bounds of fractional local metric dimension of graphs and give some characterization results. We determine the fractional local metric dimension of strong and Cartesian product of graphs.