Dominant Metric Dimension of Unit Graphs of Finite Commutative Rings

Let R be a finite commutative ring with identity and let UR denote its unit group. The unit graph GUR is the simple graph on the vertex set R in which distinct vertices x,y are adjacent if and only if x+y∈UR. A dominating resolving set is a vertex set that dominates the graph and resolves all vertices via distance representations; the minimum cardinality of such a set is the dominant metric dimension, denoted by δ⋆G. A residue–coset decomposition is developed for finite local rings R,m and is used to obtain sharp dominant-metric information directly from the quotient map R⟶R/m. For every finite local nonfield ring with residue field size q=R/m and m≥2, the exact formula δ⋆GUR=qm−1=R−R/m is proved. All minimum dominating resolving sets are characterized: optimality holds if and only if exactly one vertex is omitted from each fiber of R⟶R/m, which yields the enumeration mR/m.