A partition P of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of P. The partition dimension of G is the minimum cardinality of a locating partition of G. A pair of vertices u,v of a graph G are called twins if they have exactly the same set of neighbors other than u and v. A twin class is a maximal set of pairwise twin vertices. The twin number of a graph G is the maximum cardinality of a twin class of G.
In this paper we undertake the study of the partition dimension of a graph by also considering its twin number. This approach allows us to obtain the set of connected graphs of order n having partition dimension n-2. This set is formed by exactly 15 graphs, instead of 23, as was wrongly stated in the paper: "Discrepancies between metric dimension and partition dimension of a connected graph", published in Discrete Mathematics in 2008.
Twin-star hypothesis and cycle-free $d$-partitions of $K_{2d}$
Bowling Green State University（鲍灵格林州立大学）；Geneva College（日内瓦学院）；Institute of Mathematics of the Romanian Academy（罗马尼亚科学院数学研究所）；Columbus State University（哥伦布州立大学）
A note on partitioning the vertex set of a graph into a dominating set and a locating dominating set
Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS（克莱蒙奥弗涅大学）；University of Johannesburg（约翰内斯堡大学）；Lebanese American University（黎巴嫩美国大学）；Centrale Méditerranée, Laboratoire d’Informatique et Systèmes UMR 7020（地中海中央理工学院）；University of Turku（图尔库大学）
Quantum fractional revival on zero-divisor graphs over $\mathbb{Z}_n$
Mahidol University（玛希隆大学）；Centre of Excellence in Mathematics, MHESI（数学卓越中心，高等教育委员会）；King Mongkut’s University of Technology Thonburi (KMUTT)（蒙固皇家理工大学）
David R. Cheriton School of Computer Science, University of Waterloo（滑铁卢大学 David R. Cheriton 计算机科学学院）；Department of Combinatorics, University of Waterloo（滑铁卢大学组合数学系）
