The Vertex-Edge Resolvability of Some Wheel-Related Graphs

A vertex w ∈ V H distinguishes (or resolves) two elements (edges or vertices) a , z ∈ V H ∪ E H if d w , a ≠ d w , z . A set W m of vertices in a nontrivial connected graph H is said to be a mixed resolving set for H if every two different elements (edges and vertices) of H are distinguished by at least one vertex of W m . The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by m dim H . The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We specifically analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.