Domination Metric Dimension of Graph Classes and Its Application to Electric Logistics Network Optimization

Abstract Let G be a connected graph with vertex set V (G) and edge set E(G). A subset S ⊆ V (G) is called a dominating set of G if every vertex x ∈ V (G) \S is adjacent to at least one vertex u ∈ S. An ordered subset W ⊆ V (G) is called a resolving set of G if every pair of vertices in V (G) has a unique distance representation with respect to W. A set S ⊆ V (G) is said to be a dominating resolving set if it is both a dominating set and a resolving set. The minimum cardinality of a dominating resolving set of G is known as the domination metric dimension of G, denoted by DMD(G). In this paper, we investigate the domination metric dimension of several important graph classes, including cone graphs, corona product graphs, and tadpole graphs, and derive explicit closed-form expressions supported by rigorous mathematical proofs. Furthermore, we introduce a domination metric dimension-based framework for optimizing electric logistics networks, where strategic node selection and clustering are employed to achieve effective route optimization and distance reduction. Mathematics Subject Classification (MSC 2020): 05C12, 05C69, 90B06, 68R10.