Some Convex Polytopes Joining Paths and Their Metric Dimension

1. Introduction

Let $\mathrm{\Phi }=\mathrm{\Phi }(\theta ,\xi )$ be a fundamental, related (i.e., connected and simple) and undirected (i.e., they have no associated direction, or all of the edges are single-directional) graphical representation (or graph or network) with an edge set and a vertex set $\xi$ and $\theta$ respectively. The $metricdimension$ of the graph $\mathrm{\Phi }=\mathrm{\Phi }(\theta ,\xi )$ is the smallest number (amount) of vertices (hubs or nodes) in a set that allows a vertex to be unambiguously identified by the list of shortest paths from that vertex to those vertices. From the definition, the ordered $t$-tuple (or $t$-vector) $\eta (\nu |\zeta ):=(d_{\mathrm{\Phi }}(\nu ,{\nu }_1),d_{\mathrm{\Phi }}(\nu ,{\nu }_2),\dots ,d_{\mathrm{\Phi }}(\nu ,{\nu }_t))$ is the metric code (or metric representation) of a node $\nu$ of $\mathrm{\Phi }$ with respect to $\zeta$ in a graph $\mathrm{\Phi }$ where $\zeta =\{{\nu }_1,{\nu }_2,{\nu }_3,\dots ,{\nu }_t\}$ of nodes $\nu$ is an arranged (or ordered) subset. A set $\zeta$ that has been arranged is referred to as a resolving (locating) set of $\mathrm{\Phi }$ if every pair of different vertices of $\mathrm{\Phi }$ has a unambiguous metric representation. The cardinality (magnitude) of the subset $\zeta$, or the location number (or metric dimension) of the graph $\mathrm{\Phi }$, denoted by $dim(\mathrm{\Phi })$ or $\beta (\mathrm{\Phi })$, is the minimum size of locating set on the graph $\mathrm{\Phi }$. This is known as the metric dimension or location number of $\mathrm{\Phi }$; mathematically $\beta (\mathrm{\Phi })=dim(\mathrm{\Phi })=minimum\{|\zeta |:\zeta isresolvingset(orlocatingset)\}$. It is realized that the issue of registering this invariant is NP-hard. If a subset $\mathcal{T}$ of the arrangement of nodes $\theta (\mathrm{\Phi })$ is independent as well as resolving , then the set $\mathcal{T}$ is known as an $independentresolvingset$ for the graph $\mathrm{\Phi }$.

The $p^{th}$ coordinate (or distance component) of the code $\eta (\nu |\zeta )$ for an ordered set of vertices of $\mathrm{\Phi }$ is zero if and only if $\nu ={\nu }_{\rho }$. Consequently, it is ample to verify for any two distinct vertices $u,v\in \theta (\mathrm{\Phi })\setminus \zeta$ that $\eta (\nu |\zeta )\ne \eta (v|\zeta )$ in order to observe that the set $\zeta$ is a resolving set.

The notions of metric dimension and resolution/location trace back to the 1950s. L. M. Blumenthal [1] described them in terms of metric space. Unrelated to one another Melter and Harary [2] in 1976 and P. J. Slater [3] in 1975 introduced these concepts to graph networks. For trees, heptagonal circular ladders [4], circulant graphs [5], Harary, and other structures, the invariant metric dimension has been studied. Graph theory is a useful tool for studying the verification process in discrete science and has applications in many areas of the figure, social, and regular sciences. Applications of this invariant to problems of picture creation (or image processing) and design identification (or pattern identification) are discussed in [6]; scientific applications are presented in [7]; applications to combinatorial improvement (or optimisation) are obtained in [8]; and challenges of validation and system reveal (or network discovery) are reviewed in [9].

A family of connected graphs, or charts, let $\mho$ be formed. In particular, if $\beta (\mathrm{\Psi })=dim(\mathrm{\Psi })$ is free of the possibility of selecting of the graph $\mathrm{\Psi }$ in $\mho$ and is finite (or constrained), we assign the family $\mho$ to have stable (or constant) location number. Stated differently, $\mho$ is said to as a family with an unchanging location number if every graph in it has an indistinguishable location number [10]. Chartrand et al. showed in [7] that if a graph on $n$ nodes is a path ${\mathrm{\wp }}_n$, then it has location number one. Furthermore, for every positive integer $n$; $n\ge 3$, cycle ${\mathcal{C}}_n$ has location number two. Consequently, a collection of graphs with a constant location number is established by ${\mathcal{C}}_n$ ($n\ge 3$) and ${\mathrm{\wp }}_n$ ($n\ge 2$). In addition, the families of graphs with unchanged location numbers also include the generalised Petersen graphs $P(n,2)$ and the Harary graphs $H_{4,n}$ [5]. Recently, Sharma and Bhat studied the pane graphs and their metric dimension in [11] heptagonal circular ladder in [12,13]. Also, G. Gnecco et. al in [14] studied the optimal data collection designs in machine learning; and N. Iqbal in [15] reviewed the fractional study of the non-linear burgers’ equation.

Joining two graphs ${\mathrm{\Psi }}_1={\mathrm{\Psi }}_1({\theta }_1,{\xi }_1)$ and ${\mathrm{\Psi }}_2={\mathrm{\Psi }}_2({\theta }_2,{\xi }_2)$, denoted by $\mathrm{\Phi }={\mathrm{\Psi }}_1+{\mathrm{\Psi }}_2$, meaning a graph $\mathrm{\Phi }=\mathrm{\Phi }(\theta ,\xi )$ such that $\theta ={\theta }_1\cup {\theta }_2$ and $\xi ={\xi }_1\cup {\xi }_2\cup \{v\varsigma :v\in {\theta }_1\text{and}\varsigma \in {\theta }_2\}$. Next, we define a wheel $W_{\kappa }$ as $W_{\kappa }=K_1+{\mathcal{C}}_{\kappa }$ for $\kappa \ge 3$, a fan $F_{\kappa }=K_1+{\mathrm{\wp }}_{\kappa }$ for $\kappa \ge 1$, and a Jahangir graph $J_{2\kappa }$ ($\kappa \ge 2$) resulting from the wheel graph $W_{2\kappa }$ by deleting $\kappa$ spokes of the wheel graph (also known as a gear graph). The locating number for the fan graph $F_{\kappa }$ $(\kappa \ge 1)$, as determined by Caceres et al. in [16], is $\lfloor \frac{2\kappa +2}{5}\rfloor$ for $\kappa \notin \{1,2,3,6\}$. In [17] Chartrand et al., they determined the location number of the wheel graph $W_{\kappa }$ $(\kappa \ge 3)$, which is $\lfloor \frac{2\kappa +2}{5}\rfloor$ for $\kappa \notin \{3,6\}$. Tomescu and Javaid [18] obtained the location number of the Jahangir graph $J_{2\kappa }$ $(\kappa \ge 4)$, which is $\lfloor \frac{2\kappa }{3}\rfloor$. It should be noted that the location numbers of the plane graphs in these three families—the Fan, Wheel, and Jahangir graphs—depend on the total number of vertices in the graphs. As a result, these families do not belong to the plane graphs having fixed location numbers, also known as constant metric dimensions.

Now, a characteristic of a connected graph with respect to metric dimension equal to two. $\mathrm{\Psi }=\mathrm{\Psi }(\theta ,\xi )$ was demonstrated by Khuller et al. in [19] and is

All vertex indices in this article are assumed to be modulo $n$. The location number of the plane graph ${ϝ}_{n,m}$ is obtained in the subsequent section, and we show that $\beta ({ϝ}_{n,m})=3$ for positive integers $m$, $n$, where $m\ge 1$ and $n\ge 6$.

2. The Plane Graph ${ϝ}_{n,m}$

The plane graph ${ϝ}_{n,m}$ comprises of $n(m+5)$ number of nodes and $n(m+8)$ number of edges. It has $3n$ $5$-sided faces, and an $n$-sided face (see Figure 1). By $\xi ({ϝ}_{n,m})$ and $\theta ({ϝ}_{n,m})$, We represent the plane graph’s ${ϝ}_{n,m}$ vertex and edge ordering separately. Consequently, we have

$$\begin{array}{rlr}\theta ({ϝ}_{n,m})& =& \{{\rho }_{\tau },q_{\tau },{\ell }_{\tau },s_{\tau },{\tau }_{\tau },u_{\tau l}:1⩽\tau ⩽nand1⩽l⩽m\},\end{array}$$ and $$\begin{array}{rlr}\xi ({ϝ}_{n,m})& =& \{{\rho }_{\tau }{q}_{\tau },q_{\tau }{\ell }_{\tau },{\ell }_{\tau }{s}_{\tau },s_{\tau }{\tau }_{\tau },{\tau }_{\tau }{u}_{\tau 1},u_{\tau 1}{u}_{\tau 2},u_{\tau 2}{u}_{\tau 3},\dots ,u_{\tau m-1}{u}_{\tau m}:1\\ & & ⩽\tau ⩽n\}\cup \{{\rho }_{\tau }{\rho }_{\tau +1},{\ell }_{\tau }{q}_{\tau +1},u_{\tau 1}{\tau }_{\tau +1},s_{\tau }{s}_{\tau +1}:1⩽\tau ⩽n\}.\end{array}$$

For our motive, let us call the cycle brought forth by the arrangement of nodes $\{{\rho }_{\tau }:1⩽\tau ⩽n\}$ as the $\rho$-cycle, the cycle brought forth by the arrangement of nodes $\{q_{\tau }:1⩽\tau ⩽n\}\cup \{{\ell }_{\tau }:1⩽\tau ⩽n\}$ as the $q\ell$-cycle, the cycle that the node layout creates $\{s_{\tau }:1⩽\tau ⩽n\}$ as the $s$-cycle, the cycle that the node layout creates $\{{\tau }_{\tau }:1⩽\tau ⩽n\}\cup \{u_{\tau 1}:1⩽\tau ⩽n\}$ as the $\tau u$-cycle, and the rest of the vertices of the graphs ${ϝ}_{n,m}$ as the outer vertices. In the accompanying theorem, we discover that the location number of the plane graph, ${ϝ}_{n,m}$ is $3$ i.e., just $3$ vertices properly chosen are adequate to determine all the vertices of the plane graph ${ϝ}_{n,m}$. Note that the choice of appropriate basis node is the crux of the problem.

The desired result can alternatively be expressed as:

In the accompanying section, we acquire the location number of the plane graph ${\gimel }_{n,m}$, and for positive integers $m$, $n$ with $m\ge 1$ and $n\ge 6$ we demonstrate that $\beta ({\gimel }_{n,m})=3$.

3. The Plane Graph ${\gimel }_{n,m}$

The plane graph ${\gimel }_{n,m}$ comprises of $n(m+5)$ number of nodes and $n(m+9)$ number of edges. This graph is obtained from the graph ${ϝ}_{n,m}$ by placing the new edges between the nodes $q_{\tau }$ and $q_{\tau +1}$ $(1⩽\tau ⩽n)$. It has $2n$ $5$-sided faces, $n$ $4$ and $3$-sided faces, and an $n$-sided face (see Figure 2). By $\xi ({\gimel }_{n,m})$ and $\theta ({\gimel }_{n,m})$, we signify the arrangement of edges and vertices of the plane graph ${\gimel }_{n,m}$ separately. Consequently, we have

$$\begin{array}{rlr}\theta ({\gimel }_{n,m})& =& \{{\rho }_{\tau },q_{\tau },{\ell }_{\tau },s_{\tau },{\tau }_{\tau },u_{\tau l}:1⩽\tau ⩽nand1⩽l⩽m\},\end{array}$$ and $$\begin{array}{rlr}\xi ({\gimel }_{n,m})& =& \{{\rho }_{\tau }{q}_{\tau },q_{\tau }{\ell }_{\tau },{\ell }_{\tau }{s}_{\tau },s_{\tau }{\tau }_{\tau },{\tau }_{\tau }{u}_{\tau 1},u_{\tau 1}{u}_{\tau 2},u_{\tau 2}{u}_{\tau 3},\dots ,u_{\tau m-1}{u}_{\tau m}:1⩽\\ & & \tau ⩽n\}\cup \{{\rho }_{\tau }{\rho }_{\tau +1},q_{\tau }{q}_{\tau +1},{\ell }_{\tau }{q}_{\tau +1},u_{\tau 1}{\tau }_{\tau +1},s_{\tau }{s}_{\tau +1}:1⩽\tau ⩽n\}.\end{array}$$

For our motive, we call cycle brought forth by the arrangement of nodes $\{{\rho }_{\tau }:1⩽\tau ⩽n\}$ as the $\rho$-cycle, the cycle brought forth by the arrangement of nodes $\{q_{\tau }:1⩽\tau ⩽n\}$ as the $q$-cycle, the arrangement of nodes $\{{\ell }_{\tau }:1⩽\tau ⩽n\}$ as the set of central nodes, the cycle brought forth by the arrangement of nodes $\{s_{\tau }:1⩽\tau ⩽n\}$ as the $s$-cycle, the cycle brought forth by the arrangement of vertices $\{{\tau }_{\tau }:1⩽\tau ⩽n\}\cup \{u_{\tau 1}:1⩽\tau ⩽n\}$ as the $\tau u$-cycle, and the rest of the vertices of the graphs ${\gimel }_{n,m}$ as the outer vertices. In the accompanying theorem, we discover that the location number of the plane graph, ${\gimel }_{n,m}$ is $3$ i.e., just $3$ vertices properly chosen are adequate to determine all the vertices of the plane graph ${\gimel }_{n,m}$. Note that the choice of appropriate basis node is the crux of the problem.