The partition dimension of a graph

Abstract

For a vertex v of a connected graph G and a subset S of V (G), the distance between v and S is dIt is shown that the partition dimension of a graph G is bounded above by 1 more than its metric dimension. An upper bound for the partition dimension of a bipartite graph G is given in terms of the cardinalities of its partite sets, and it is shown that the bound is attained if and only if G is a complete bipartite graph. Graphs of order n having partition dimension 2, n, or n − 1 are characterized.Mathematics Subject Classification (1991). 05C12.

On The Partition Dimension of Cm + Pn Graph

“…In 2000, Chartrand et al [2] determined the partition dimension of some graph classes which result pd(G) = 2 if and only if G = P n , pd(G) = n if and only if G = K n , and pd(G) = 3 if G = C n for n ≥ 3. In 2007, Tomescu et al [5] formed the partition dimension of a wheel graph.…”

“…One of the interesting concepts in graph theory is so called the partition dimension. The partition dimension of a graph was introduced by Chartrand et al [1] in 1998. The partition dimension is a variation of the metric dimension introduced by Slater in 1975 (Chartrand et al [3]).…”

On The Partition Dimension of Disconnected Graphs

“…Chartrand, et al [1] in 1998 introduced the idea for the partition dimension of connected graphs. This concept is a variant of the metric dimension of a graph described independently by Slater [2] in 1975 and by Harary & Melter [3] in 1976.…”

“…In [4], Chartrand, et al established a relation between the partition dimension and the metric dimension of a graph. They also proved that the only graph of order 2 with the partition dimension two is a path , while the only graph with the partition dimension is a complete graph .…”

On the star partition dimension of comb product of cycle and complete graph

“…In his paper, Slater called this concept as a locating set. Chartrand et.al [5] introduced the concept of partition dimension of G. They introduced the same concept of resolving partition with a partition dimension of graphs. For S ⊆ V (G) with vertex v ∈ V (G), the distance between v and S is d(v, S) = min{d(v, x)}.…”