Abstract
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.
Highlights
Is known as metric representation of i with respect to R
We can affirm that Π is the resolving partition of the graph G if different vertices have unique partition representation, that is, r(i|Π) ≠ r(j|Π), where i, j ∈ VG. e partition dimension of the graph G is than the minimum number of resolving partition set in G
The graph with partition dimension |V| − 3 is discussed [6] and the graph obtained by graph operations and its corresponding partition dimension is studied in [7]. e bounds on the partition dimension for convex polytopes are studied in [8,9,10] and bounds of partition on the circulant and multipartite discussed in [11, 12]. e partition dimension of chemical structure fullerenes graphs is studied in [13], and on the bounded partition, dimension of the Cartesian product of graphs are studied in [14]
Summary
Is known as metric representation of i with respect to R. We consider the upper bound for the partition dimension of the generalized Petersen graph in terms of the cardinalities of its partite sets. E study on the resolving set and metric dimension of Petersen and generalized
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