On Constant Metric Dimension of Some Generalized Convex Polytopes (original) (raw)

Computation of Metric Dimension of Certain Subdivided Convex Polytopes

Journal of Mathematics, 2022

The distance d z 1 , z 2 from vertex z 1 ∈ V G to z 2 ∈ V G is minimum length of z 1 , z 2 -path in a given connected graph G having E(G) and V(G) edges and vertices’/nodes’ sets, respectively. Suppose Z = z 1 , z 2 , z 3 , … , z m ⊆ V G is an order set and c ∈ V G , and the code of c with reference to Z is the m-tuple {d(c, z1), d(c, z2), d(c, z13), …, d(c, zk)}. Then, Z is named as the locating set or resolving set if each node of G has unique code. A locating set of least cardinality is described as a basis set for the graph G , and its cardinal number is referred to as metric dimension symbolized by dim G . Metric dimension of certain subdivided convex polytopes S T n has been computed, and it is concluded that just four vertices are sufficient for unique coding of all nodes belonging to this family of convex polytopes.

On the metric dimension of rotationally-symmetric convex polytopes

Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let FFF be a family of connected graphs Gn:F=(Gn)n≥1G_n : F = (G_n)_n ≥ 1Gn:F=(Gn)n≥1 depending on nnn as follows: the order ∣V(G)∣=ϕ(n)|V (G)| = ϕ(n)∣V(G)∣=ϕ(n) and limn→∞ϕ(n)=∞lim n→∞ ϕ(n) = ∞limn→∞ϕ(n)=∞. If there exists a constant C>0C > 0C>0 such that dim(Gn)≤Cdim(G_n) ≤ Cdim(Gn)≤C for every n≥1n ≥ 1n≥1 then we shall say that F has bounded metric dimension, otherwise F has unbounded metric dimension. If all graphs in FFF have the same metric dimension, then FFF is called a family of graphs with constant metric dimension. In this paper, we study the metric dimension of some classes of convex polytopes which are rotationally-symmetric. It is shown that these classes of convex polytoes have the constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension.

Computation of the Double Metric Dimension in Convex Polytopes

Journal of Mathematics, 2021

A source detection problem in complex networks has been studied widely. Source localization has much importance in order to model many real-world phenomena, for instance, spreading of a virus in a computer network, epidemics in human beings, and rumor spreading on the internet. A source localization problem is to identify a node in the network that gives the best description of the observed diffusion. For this purpose, we select a subset of nodes with least size such that the source can be uniquely located. This is equivalent to find the minimal doubly resolving set of a network. In this article, we have computed the double metric dimension of convex polytopes R n and Q n by describing their minimal doubly resolving sets.

Computing the edge metric dimension of convex polytopes related graphs

Journal of Mathematics and Computer Science

Let G = (V(G), E(G)) be a connected graph and d(f, y) denotes the distance between edge f and vertex y, which is defined as d(f, y) = min{d(p, y), d(q, y)}, where f = pq. A subset W E ⊆ V(G) is called an edge metric generator for graph G if for every two distinct edges f 1 , f 2 ∈ E(G), there exists a vertex y ∈ W E such that d(f 1 , y) = d(f 2 , y). An edge metric generator with minimum number of vertices is called an edge metric basis for graph G and the cardinality of an edge metric basis is called the edge metric dimension represented by edim(G). In this paper, we study the edge metric dimension of flower graph f n×3 and also calculate the edge metric dimension of the prism related graphs D n and D t n. It has been concluded that the edge metric dimension of D n is bounded, while of f n×3 and D t n is unbounded.

COMPUTING THE METRIC DIMENSION OF CONVEX POLYTOPES GENERATED BY WHEEL RELATED GRAPHS

An ordered set W = {w1,. .. , w k } V (G) of vertices of G is called a resolving set or locating set for G if every vertex is uniquely determined by its vector of distances to the vertices in W. A resolving set of minimum cardi-nality is called a metric basis for G and this cardinality is the metric dimension or location number of G, denoted by β(G). The metric dimension of certain wheel related graphs has been studied recently in [22]. In this paper, we extend this study to infinite classes of convex polytopes generated by wheel related graphs. We prove that these infinite classes of convex polytopes generated by wheel related graphs have unbounded metric dimension. It is natural to ask for the characterization of graphs with unbounded metric dimension.

Fault-Tolerant Metric Dimension of Generalized Wheels and Convex Polytopes

Mathematical Problems in Engineering, 2020

For a graph G , an ordered set S ⊆ V G is called the resolving set of G , if the vector of distances to the vertices in S is distinct for every v ∈ V G . The minimum cardinality of S is termed as the metric dimension of G . S is called a fault-tolerant resolving set (FTRS) for G , if S \ v is still the resolving set ∀ v ∈ V G . The minimum cardinality of such a set is the fault-tolerant metric dimension (FTMD) of G . Due to enormous application in science such as mathematics and computer, the notion of the resolving set is being widely studied. In the present article, we focus on determining the FTMD of a generalized wheel graph. Moreover, a formula is developed for FTMD of a wheel and generalized wheels. Recently, some bounds of the FTMD of some of the convex polytopes have been computed, but here we come up with the exact values of the FTMD of two families of convex polytopes denoted as D k for k ≥ 4 and Q k for k ≥ 6 . We prove that these families of convex polytopes have constan...

Minimal doubly resolving sets and the strong metric dimension of some convex polytopes

Applied Mathematics and Computation, 2012

We consider the problem of determining the cardinality ψ(H 2,k) of minimal doubly resolving sets of Hamming graphs H 2,k. We prove that for k ≥ 6 every minimal resolving set of H 2,k is also a doubly resolving set, and, consequently, ψ(H 2,k) is equal to the metric dimension of H 2,k , which is known from the literature. Moreover, we find an explicit expression for the strong metric dimension of all Hamming graphs H n,k .

Computation of certain measures of proximity between convex polytopes: A complexity viewpoint

1992

Four problems of proximity between two convex polytopes in R' are considered. The convex polytopes are represented as convex hulls of finite sets of points. Let the total number of points in the two finite sets be n. W e show that three of the proximity problems, viz., checking intersection, checking whether the polytopes are just touching and finding the distance between them, can be solved in O(n) time for fixed s and in polynomial time f o r va y i n g s. We also show that the fourth proximity problem of finding the intensity of collision f o r varying s is NP-complete.

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.

Graphic vertices of the metric polytope

Discrete Mathematics, 1996

The metric polytope ,//~. is defined by the triangle inequalities: xij-Xik-Xjk <~ 0 and xii + Xik + Xjk ~< 2 for all triples i,j, k of(1 ..... n}. The integral vertices of ~¢¢~. are the incidence vectors of the cuts of the complete graph Kn. Therefore, ~¢~. is a relaxation of the cut polytope of K~. We study here the fractional vertices of ~¢¢~. Many of them are constructed from graphs; this is the case for the one-third-integral vertices. One-third-integral vertices are, in a sense, the simplest fractional vertices of ~'~, as ./t'~ has no half-integral vertices. Several constructions for one-third-integral vertices are presented. In particular, the graphic vertices arising from the suspension of a tree are characterized. We describe the symmetries of ~¢/~. and obtain that the vertices are partitioned into switching classes. With the exception of the cuts which are pairwise adjacent, it is shown that no two vertices of the same switching class are adjacent on ~¢~n. The question of adjacency of the fractional vertices to the integral ones is also addressed. All the vertices of ,//~. for n ~< 6 are described.