Dominant Metric Dimension of Star Fan Graphs

A dominating set of a connected graph G = (V, E) is a subset D of V (G) such that every vertex of G is either in D or adjacent to a vertex in D . A resolving set of a connected graph G is a subset D of V (G) such that each vertex v of G has a unique representation with respect to D . A dominant resolving set of a graph G is a subset D of V (G) that resolves all vertices of G and dominates G . The dominant metric dimension, Ddim(G) , is the cardinality of the smallest such set. The dominant metric dimension of a graph combines metric dimension (location identification) with domination (coverage), making it ideal for sensor networks that require both unique positioning and full coverage. Similarly, a dominant edge resolving set of a connected graph G is a vertex subset D of V that resolves all edges and is a covering of G. Star fan graphs are composite graph structures formed by attaching fan graphs to the pendant vertices of a star graph, creating hierarchical networks that are useful for modeling hub-and-spoke systems with additional path structures. This paper computes the dominant metric dimension and dominant edge metric dimension of star fan graphs.