The Partition Dimension of a Path Graph

Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph and is a subset of writen . Suppose there is , then the distance between and is denoted in the form . There is an ordered set of -partitions of, writen then the representation of with respect tois the The set of partitions ofis called a resolving partition if the representation of each to is different. The minimum cardinality of the solving-partition to is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitions of path graph, with , and are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely pd (Pn)=2Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph and is a subset of writen . Suppose there is , then the distance between and is denoted in the form . There is an ordered set of -partitions of, writen then the representation of with respect tois the The set of partitions ofis called a resolving partition if the representation of each to is different. The minimum cardinality of the solving-partition to is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitionsof path graph, with , and are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely.

On the Locating Chromatic Number of Barbell Shadow Path Graph

The locating-chromatic number was introduced by Chartrand in 2002. The locating chromatic number of a graph is a combined concept between the coloring and partition dimension of a graph. The locating chromatic number of a graph is defined as the cardinality of the minimum color classes of the graph. In this paper, we discuss about the locating-chromatic number of shadow path graph and barbell graph containing shadow graph.

Fault-Tolerant Partition Resolvability of Cyclic Networks

Graph invariants provide an amazing tool to analyze the abstract structures of networks. The interaction and interconnection between devices, sensors, and service providers have opened the door for an eruption of mobile over the web applications. Structure of web sites containing number of pages can be represented using graph, where web pages are considered to be the vertices, and an edge is a link between two pages. Figuring resolving partition of the graph is an intriguing inquest in graph theory as it has many applications such as sensor design, compound classification in chemistry, robotic navigation, and Internet network. The partition dimension is a graph parameter akin to the concept of metric dimension, and fault-tolerant partition dimension is an advancement in the line of research of partition dimension of the graph. In this paper, we compute fault-tolerant partition dimension of alternate triangular cycle, mirror graph, and tortoise graphs.

On Fault-Tolerant Partition Dimension of Homogeneous Caterpillar Graphs

Metric-related parameters in graph theory have several applications in robotics, navigation, and chemical strata. An important such parameter is the partition dimension of graphs that plays an important role in engineering, computer science, and chemistry. In the context of chemical and pharmaceutical engineering, these parameters are used for unique representation of chemical compounds and their structural analysis. The structure of benzenoid hydrocarbon molecules is represented in the form of caterpillar trees and studied for various attributes including UV absorption spectrum, molecular susceptibility, anisotropy, and heat of atomization. Several classes of trees have been studied for partition dimension; however, in this regard, the advanced variant, the fault-tolerant partition dimension, remains to be explored. In this paper, we computed fault-tolerant partition dimension for homogeneous caterpillars C p ; 1 , C p ; 2 , and C p ; 3 for p ≥ 5 , p ≥ 3 , and p ≥ 4 , respectively, and it is found to be constant. Further numerical examples and an application are furnished to elaborate the accuracy and significance of the work.

Partition Dimension of Generalized Petersen Graph

Let G = V G , E G be the connected graph. For any vertex i ∈ V G and a subset B ⊆ V G , the distance between i and B is d i ; B = min d i , j | j ∈ B . The ordered k -partition of V G is Π = B 1 , B 2 , … , B k . The representation of vertex i with respect to Π is the k -vector, that is, r i | Π = d i , B 1 , d i , B 2 , … , d i , B k . The partition Π is called the resolving (distinguishing) partition if r i | Π ≠ r j | Π , for all distinct i , j ∈ V G . The minimum cardinality of the resolving partition is called the partition dimension, denoted as pd G . In this paper, we consider the upper bound for the partition dimension of the generalized Petersen graph in terms of the cardinalities of its partite sets.

The Locating-Chromatic Number of Origami Graphs

The locating-chromatic number of a graph combines two graph concepts, namely coloring vertices and partition dimension of a graph. The locating-chromatic number is the smallest k such that G has a locating k-coloring, denoted by χL(G). This article proposes a procedure for obtaining a locating-chromatic number for an origami graph and its subdivision (one vertex on an outer edge) through two theorems with proofs.