A novel resolvability parameter: Face metric dimension and its applications

Faces in graphs play a crucial role in understanding the structural properties of planar graphs.They represent the regions or bounded areas formed by the edges of the graph when it is embedded in the plane.The concept of faces provides insights into the connectivity and layout of systems, helping analyze the geometry and topology of networks, communication systems, and various real-world applications.In graph theory, the concept of resolvability plays a significant role in identifying distinct elements within a graph based on distances.In graph theory, the concept of resolvability plays a significant role in identifying distinct elements within a graph based on distances.Let G be a connected planar graph with vertex V(G), edge set E(G), and face set F(G).The distance between a face f and a vertex v is defined as the minimum distance from v to any vertex incident to f .In this work, we introduce a new resolvability parameter for connected planar graphs, referred to as the face metric dimension.A face-resolving set R V(G) is a set of vertices such that for every pair of distinct faces f 1 , f 2 F(G), there exists at least one vertex r R for which the distances d( f 1 , r) and d( f 2 , r) are distinct.The face metric dimension of G, denoted f md(G), is the minimum cardinality of a face-resolving set.This new metric provides insight into the structure of planar graphs and offers a novel perspective on the analysis of graph resolvability.