Structure theory for a characterization of the metric dimension of graphs

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.