@article{Luo2026, 
author = {Haichang Luo and Ghulam Haidar and Murad ul Islam Khan and Sakander Hayat and Mohammed J. F. Alenazi},
title = {Structure theory for a characterization of the metric dimension of graphs},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {3},
pages = {6019-6029},
keywords = {graph theory, resolvability, metric dimension, extreme value},
url = {https://www.sciopen.com/article/10.3934/math.2026249},
doi = {10.3934/math.2026249},
abstract = {Characterizing simple connected graphs of order    n having metric dimension    n  −  2, solved by Chartrand et al. [Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105  (2000), 99-113], is a foundational result in the study of metric dimension. This article presents a refined proof for the non-bipartite case of the original theorem. While this work does not present a new characterization result, its primary contribution is methodological: We reframe the original's lengthy case-by-case elimination argument as a series of standalone lemmas, which we use to formally establish the structural properties that such a graph must satisfy. Building upon these properties, we then provide a direct, constructive proof demonstrating that the graph structure is necessarily the join of a complete and empty graph. This method offers a more elegant argument for this important characterization and also provides a clearer understanding of why this specific graph family emerges.}
}