Metric Dimension Of Graph Research Articles

Metric Dimension of Corona Product and Join Graph of Zero Divisor Graphs of Direct Product of Finite Fields

The metric dimension of a graph is the smallest number of vertices such that all vertices are uniquely determined by their distances to the chosen vertices. The corona product of graphs G and H is the graph G ⊙ H obtained by taking one copy of G , called the center graph, | V ( G ) | copies of H , called the outer graph, and making the j t h vertex of G adjacent to every vertex of the j t h copy of H , where 1 ⩽ j ⩽ | V ( G ) | . The Join graph G + H of two graphs G and H is the graph with vertex set V ( G + H ) = V ( G ) ∪ V ( H ) and edge set E ( G + H ) = E ( G ) ∪ E ( H ) ∪ { u v : u ∈ V ( G ) , v ∈ V ( H ) } . In this paper, we determine the Metric dimension of Corona product and Join graph of zero divisor graphs of direct product of finite fields.

Edge Metric Dimension of Comb Product Cycles Versus Stars and Fans

The edge metric dimension finding of a graph resulting from the comb product on a cycle graph to the complete graph and the fact about the dominant vertex in the complete graph arise a problem about the edge metric dimension of a graph resulting from the comb product on a cycle graph to the other graphs that have dominant vertex such as star and fan graph. This study aims to determine the edge metric dimension of the graph resulting from the comb product on a cycle graph to the star and fan graph, respectively. The results show that the upper and lower bound of the edge metric dimension of both graphs are equivalent so that an exact edge metric dimension is obtained. The edge metric dimension of the graph comb product of the cycle graph C_n to the star graph S_(1,m) is n(m-1). The edge metric dimension of the graph comb product of the cycle graph C_n to the fan graph F_(1,m) is n(m-1).

A special graph for the connected metric dimension of graphs

Given a connected graph G=(V, E), let d(x, y) represent the separation between x and y at its vertices. If each vertex in a collection B is uniquely identified by its vector of distances to the vertices in B, then that set of vertices resolves a graph G. A metric dimension of G is represented by dim(G) and is the smallest cardinality of a resolving set of G. If the subgraph B- induced by B is a nontrivial connected subgraph of G, then a resolving set B of G is connected. The metric dimension of G is the cardinality of the minimal resolving set, while the connected metric dimension of G is the cardinality of the smallest connected resolving set. The connected metric dimension of the knots graph, whitehead link graph and jewel graph are determined in this study. Finally, we derive the explicit formulas for the triangular book graph, quadrilateral book graph and crystal planar map.

Metric dimension of unit graphs

The notion of the metric dimension of a graph is well-known and its study is well entrenched in the literature. In this paper, we introduce new classes of graphs that exhibit remarkable characteristic: the metric dimension and the diameter of the unit graph are equal. Additionally, we provide a characterization of finite commutative rings [Formula: see text], wherein the metric dimension assumes a value [Formula: see text], where [Formula: see text]. Further, we also demonstrate an exhaustive examination that ascertains the precise finite commutative rings [Formula: see text], in which domination number is equal to metric dimension of unit graph.

Dominant Mixed Metric Dimension of Graph

For $k-$ordered set $W=\{s_1, s_2,\dots, s_k \}$ of vertex set $G$, the representation of a vertex or edge $a$ of $G$ with respect to $W$ is $r(a|W)=(d(a,s_1), d(a,s_2),\dots, d(a,s_k))$ where $a$ is vertex so that $d(a,s_i)$ is a distance between of the vertex $v$ and the vertices in $W$ and $a=uv$ is edge so that $d(a,s_i)=min\{d(u,s_i),d(v,s_i)\}$. The set $W$ is a mixed resolving set of $G$ if $r(a|W)\neq r(b|W)$ for every pair $a,b$ of distinct vertices or edge of $G$. The minimum mixed resolving set $W$ is a mixed basis of $G$. If $G$ has a mixed basis, then its cardinality is called mixed metric dimension, denoted by $dim_m(G)$. A set $W$ of vertices in $G$ is a dominating set for $G$ if every vertex of $G$ that is not in $W$ is adjacent to some vertex of $W$. The minimum cardinality of dominating set is domination number , denoted by $\gamma(G)$. A vertex set of some vertices in $G$ that is both mixed resolving and dominating set is a mixed resolving dominating set. The minimum cardinality of mixed resolving dominating set is called dominant mixed metric dimension, denoted by $\gamma_{mr}(G)$. In our paper, we will investigated the establish sharp bounds of the dominant mixed metric dimension of $G$ and determine the exact value of some family graphs.

On the metric dimension of graphs associated with irreducible and Arf numerical semigroups

A subset Δ of non-negative integers N 0 is called a numerical semigroup if it is a submonoid of N 0 and has a finite complement in N 0 . A graph G Δ is called a Δ ( α , β ) -graph if there exists a numerical semigroup Δ with multiplicity α and embedding dimension β such that V ( G Δ ) = { v i : i ∈ N 0 ∖ Δ } and E ( G Δ ) = { v i v j ⇔ i + j ∈ Δ } . In this article, we compute the Δ ( α , β ) -graphs for irreducible and Arf numerical semigroups having a metric dimension of 2. It is proved that if Δ be an irreducible and arf numerical semigroup then there are exactly 2 and 8 non-isomorphic Δ ( α , β ) -graphs respectively, whose metric dimension is 2.

Binary rat swarm optimizer algorithm for computing independent domination metric dimension problem

In this article, we look at the NP-hard problem of determining the minimum independent 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 recognized by its vector of distances to the vertices in B. If there are no neighboring vertices in a resolving set B of G, then B is independent. Every vertex of G that does not belong to B must be a neighbor of at least one vertex in B for a resolving set to be dominant. The metric dimension of G, independent metric dimension of G, and independent dominant metric dimension of G are, respectively, the cardinality of the smallest resolving set of G, the minimal independent resolving set, and the minimal independent domination resolving set. We propose the first attempt to use a binary version of the Rat Swarm Optimizer Algorithm (BRSOA) to heuristically calculate the smallest independent dominant resolving set of graphs. The search agent of BRSOA are binary-encoded and used to identify which one of the vertices of the graph belongs to the independent domination resolving set. The feasibility is enforced by repairing search agent such that an additional vertex created from vertices of G is added to B, and this repairing process is repeated until B becomes the independent domination resolving set. Using theoretically computed graph findings and comparisons to competing methods, the proposed BRSOA is put to the test. BRSOA surpasses the binary Grey Wolf Optimizer (BGWO), the binary Particle Swarm Optimizer (BPSO), the binary Whale Optimizer (BWOA), the binary Gravitational Search Algorithm (BGSA), and the binary Moth-Flame Optimization (BMFO), according to computational results and their analysis.

Connected metric dimension of the class of ladder graphs

Numerous applications, like robot navigation, network verification and discovery, geographical routing protocols, and combinatorial optimization, make use of the metric dimension and connected metric dimension of graphs. In this work, the connected metric dimension types of ladder graphs, namely, ladder, circular, open, and triangular ladder graphs, as well as open diagonal and slanting ladder graphs, are studied.

Bi-Edge Metric Dimension of Graphs

Bi-Edge Metric Dimension of Graphs

On the metric-based resolving parameter of the line graph of certain structures

Let G be a graph and R = {r1, r2, …, rk} be an ordered subset of vertices of G, if every two vertices of G have different representation r (v|R) = (d (v, r1) , d (v, r2) , …, d (v, rk)) with respect to R, then R is said to be a metric-based resolving parameter or resolving set of G and its minimum cardinality is called the metric dimension of graph G . Metric dimension is considered as an important applied concept of graph theory especially in the localization of a network and also in the chemical graph theoretical study of molecular compounds. Therefore, it is hot topic to study for different families of graphs as well. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In this paper, we determine the metric-based resolving parameter of line graph of a convex polytope Sn, and conclude that it has constant metric dimension but vary with the parity of n . This article presents a measurement of the line graph of a convex polytope, denoted as ( S n ) . The subsequent section provides the metric dimension of the resulting graph. There are two scenarios pertaining to the metric dimension of a selected graph with respect to the metric dimension. The metric dimension of even cycle-based convex polytopes is three, whereas for other values, the metric dimension is four.

On the Local Metric Dimension of Graphs

Let [Formula: see text] be a graph. A set [Formula: see text] is a local resolving set of [Formula: see text] if there exists [Formula: see text] such that [Formula: see text] for any [Formula: see text]. The local metric dimension [Formula: see text] of [Formula: see text] is the minimum cardinality of all the local resolving sets of [Formula: see text]. In this paper, we characterize the graphs with [Formula: see text]. Next, we obtain the Nordhaus–Gaddum-type results for local metric dimension. Finally, the local metric dimension of several graph classes is given.

On metric dimension of hendecagonal circular ladder $H_{n}$

Let $\zeta=(V,E)$ be a $n$th order connected graph. If the distance vectors to the vertices in an ordered subset $G$ of vertices can uniquely identify each vertex of the graph $\zeta$, then the set $G$ is known as resolving set for the graph $\zeta$. The resolving set $G$ with smallest cardinality serves as the metric dimension of graph $\zeta$ and this resolving set serves as the metric basis for $\zeta$. In this article, two families of convex polytopes that are closely linked are demonstrated and it is found that the metric dimension is three for both the families.

Multi Polar q-Rung Orthopair Fuzzy Graphs with Some Topological Indices

The importance of symmetry in graph theory has always been significant, but in recent years, it has become much more so in a number of subfields, including but not limited to domination theory, topological indices, Gromov hyperbolic graphs, and the metric dimension of graphs. The purpose of this monograph is to initiate the idea of a multi polar q-rung orthopair fuzzy graphs (m-PqROPFG) as a fusion of multi polar fuzzy graphs and q-rung orthopair fuzzy graphs. Moreover, for a vertex of multi polar q-rung orthopair fuzzy graphs, the degree and total degree of the vertex are defined. Then, some product operations, inclusive of direct, Cartesian, semi strong, strong lexicographic products, and the union of multi polar q-rung orthopair fuzzy graphs (m-PqROPFGs), are obtained. Also, at first we define some degree based fuzzy topological indices of m-PqROPFG. Then, we compute Zareb indices of the first and second kind, Randic indices, and harmonic index of a m-PqROPFG.

Treelength of series–parallel graphs

The length of a tree-decomposition of a graph is the maximum distance (in the graph) between two vertices of a same bag of the decomposition. The treelength of a graph is the minimum length among its tree-decompositions. Treelength of graphs has been studied for its algorithmic applications in classical metric problems such as Traveling Salesman Problem or metric dimension of graphs and also, in compact routing in the context of distributed computing. Deciding whether the treelength of a general graph is at most 2 is NP-complete (graphs of treelength one are precisely the chordal graphs), and it is known that the treelength of a graph cannot be approximated up to a factor less than 32 (the best known approximation algorithm for treelength has an approximation ratio of 3). However, nothing is known on the computational complexity of treelength in planar graphs, except that the treelength of any outerplanar graph is equal to the third of the size of a largest isometric cycle. This work initiates the study of treelength in planar graphs by considering the next natural subclass of planar graphs, namely the one of series–parallel graphs.We first fully describe the treelength of melon graphs (set of pairwise internally disjoint paths linking two vertices), showing that, even in such a restricted graph class, the expression of the treelength is not trivial. Then, we show that treelength can be approximated up to a factor 32 in series–parallel graphs. Our main result is a quadratic-time algorithm for deciding whether a series–parallel graph has treelength at most 2. Our latter result relies on a characterization of series–parallel graphs with treelength 2 in terms of an infinite family of forbidden isometric subgraphs.

A HYBRID OPTIMIZATION ALGORITHMS FOR SOLVING METRIC DIMENSION PROBLEM

In this paper, we consider the NP-hard problem of finding the minimum resolving set of graphs. A vertex set B of a connected graph G resolves G if every vertex of G is uniquely identified by its vector of distances to the vertices in B. The cardinality of the minimal resolving set is the metric dimension of G. The metric dimension appears in various fields such as network discovery and verification, robot navigation, combinatorial optimization and pharmaceutical chemistry, etc. In this study, we introduce a hybrid approach (WCA_WOA) for computing the metric dimension of graphs that combines the water cycle algorithm and a whale optimisation algorithm. The WOA algorithm hybridises the WCA in order to obtain the optimal result and manage the optimization process. The results of the experiments show that the WCA_WOA hybrid algorithm outperforms the WCA, WOA, and particle swarm optimization methods

On the Dominant Local Resolving Set of Vertex Amalgamation 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

The localization number and metric dimension of graphs of diameter 2

We consider the localization number and metric dimension of certain graphs of diameter $2$, focusing on families of Kneser graphs and graphs without 4-cycles. For the Kneser graphs with a diameter of $2$, we find upper and lower bounds for the localization number and metric dimension, and in many cases these parameters differ only by an additive constant. Our results on the metric dimension of Kneser graphs improve on earlier ones, yielding exact values in infinitely many cases. We determine bounds on the localization number and metric dimension of Moore graphs of diameter $2$ and polarity graphs.

The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees

Consider a robot that is navigating in a space represented by a graph and wants to know its current location. It can send a signal to find out how far it is from each set of fixed landmarks. We study the problem of computing the minimum number of landmarks required, and where they should be placed so that the robot can always determine its location. The set of nodes where the landmarks are located is called the metric basis of the graph, and the number of landmarks is called the metric dimension of the graph. On the other hand, the metric dimension of a graph <I>G</I> is the smallest size of a set <I>B</I> of vertices that can distinguish each vertex pair of <I>G</I> by the shortest-path distance to some vertex in <I>B</I>. The finding of the metric dimension of an arbitrary graph is an NP-complete problem. Also, the metric dimension has several applications in different areas, such as geographical routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we study the metric dimension of subdivisions of several graphs, including the Lilly graph, the Tadpole graph, and the special trees star tree, bistar tree, and coconut tree.

Nonlocal Metric Dimension of Graphs

Nonlocal metric dimension $$\text {dim}_{\textrm{n}\ell } (G)$$ of a graph G is introduced as the cardinality of a smallest nonlocal resolving set, that is, a set of vertices which resolves each pair of non-adjacent vertices of G. Graphs G with $$\text {dim}_{\textrm{n}\ell }(G) = 1$$ or with $$\text {dim}_{\textrm{n}\ell }(G) = n(G)-2$$ are characterized. The nonlocal metric dimension is determined for block graphs, for corona products, and for wheels. Two upper bounds on the nonlocal metric dimension are proved. An embedding of an arbitrary graph into a supergraph with a small nonlocal metric dimension and small diameter is presented.

On a Conjecture About the Local Metric Dimension of Graphs

On a Conjecture About the Local Metric Dimension of Graphs