Bounds on the Partition Dimension of Convex Polytopes

Abstract

Aims and objective: The idea of partition and resolving sets play an important role in various areas of engineering, chemistry and computer science such as robot navigation, facility location, pharmaceutical chemistry, combinatorial optimization, networking, and mastermind game.

Methods: In a graph, to obtain the exact location of a required vertex, which is unique from all the vertices, several vertices are selected; this is called resolving set, and its generalization is called resolving partition, where selected vertices are in the form of subsets. A minimum number of partitions of the vertices into sets is called partition dimension.

Results: It was proved that determining the partition dimension of a graph is a nondeterministic polynomial time (NP) problem. In this article, we find the partition dimension of convex polytopes and provide their bounds.

Conclusion: The major contribution of this article is that due to the complexity of computing the exact partition dimension, we provide the bounds and show that all the graphs discussed in the results have partition dimensions either less or equals to 4, but not greater than 4.

Keywords: Partition dimension; bounded partition dimension.; convex polytopes; resolving partition; resolving sets.