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 incidence 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 \( R \subseteq V(G) \) is a set of vertices such that for every pair of distinct faces \( f_1, f_2 \in F(G) \), there exists at least one vertex \( r \in R \) for which the distances \( d(f_1, r) \) and \( d(f_2, r) \) are distinct. The face metric dimension of \( G \), denoted \( \ fmd(G) \), 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.
A finite, connected simple graph \(G\) is a geodetic graph if and only if for each pair of vertices \(v_i, v_j\) there exists a unique distance path (or unique shortest \(v_iv_j\)-path). The insertion of vertices in an edge or edges of a non-geodetic graph \(G\) to, if possible, obtain a resultant geodetic graph is called geodetication of the graph \(G\). The paper introduces two new graph parameters generally called the Ruv\(\acute{e}\) numbers of a graph. The Ruv\(\acute{e}\) numbers of \(G\) are denoted by \(\rho_1(G)\) and \(\rho_2(G)\) respectively, and \(\rho_1(G) = \rho_2(G) = 0\) if and only if \(G\) is geodetic. Furthermore, for some graphs the parameter, \(\rho_1(G) \to \infty\). The latter graphs \(G\) do not permit geodetication in respect of \(\rho_1(G)\). It is evident that geodetication presents various challenging minimization problems. The core field of application will be, restricting graphs to distance path uniqueness. Intuitive applications are foreseen in military science, IT anti-hacking coding and predictive flow through networks.
The anchored Dyck words of length \(n=2k+1\) (obtained by prefixing a 0-bit to each Dyck word of length \(2k\) and used to reinterpret the Hamilton cycles in the odd graph \(O_k\) and the middle-levels graph \(M_k\) found by M\”utze et al.) represent in \(O_k\) (resp., \(M_k\)) the cycles of an \(n\)- (resp., \(2n\)-) 2-factor and its cyclic (resp., dihedral) vertex classes, and are equivalent to Dyck-nest signatures. A sequence is obtained by updating these signatures according to the depth-first order of a tree of restricted growth strings (RGS’s), reducing the RGS-generation of Dyck words by collapsing to a single update the time-consuming \(i\)-nested castling used to reach each non-root Dyck word or Dyck nest. This update is universal, for it does not depend on \(k\).
