School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601,China.
Department of Mathematics, COMSATS University Islamabad Lahore Campus, Lahore ,Pakistan.
Comb Chem High Throughput Screen. 2022;25(3):547-553. doi: 10.2174/1386207323666201204144422.
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.
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.
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.
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.
划分和解析集的概念在机器人导航、设施定位、药物化学、组合优化、网络和国际象棋游戏等工程、化学和计算机科学的各个领域中都起着重要作用。
在图中，为了从所有顶点中找到所需的唯一顶点的精确位置，选择几个顶点；这被称为解析集，其推广称为解析分区，其中选择的顶点是子集的形式。将顶点划分为集合的最小数量称为分区维数。
证明了确定图的分区维数是一个非确定性多项式时间（NP）问题。在本文中，我们找到了凸多面体的分区维数，并提供了它们的界。
本文的主要贡献在于，由于计算精确分区维数的复杂性，我们提供了界限，并表明结果中讨论的所有图的分区维数要么小于或等于 4，要么大于 4。
