Computation of mixed resolvability for a circular ladder and its unbounded nature

Abstract

Let Γ = Γ(V ,E) be a simple, planar, connected, and undirected graph. The article primarily concentrates on a category of planar graphs, detailing the explicit identification of each member within this graph family. Within the domain of graph theory, the parameters used to uniquely identify vertices and edges of a graph are commonly referred to as variants of metric dimension, collectively known as resolvability parameters. The present study focuses on the intricate planar structure of a five-sided circular ladder (pentagonal); denoted by Ph5, and investigate some of the recently introduced resolvability parameters for it, which are mixed metric basis and mixed metric dimension. We prove that the mixed metric dimension for Ph5 is unbounded, and it depends upon the number of vertices present in it. The comparison between several resolvability parameters, viz., metric dimension and edge metric dimension, for Ph5 with mixed metric dimension have also been incorporated in this manuscript, indicating higher level of complexity for resolving both edge and vertex-based relationships. Moreover, several theoretical as well as application based properties, including examples, have also been discussed for Ph5.