The true twin classes-based investigation for connected local dimensions of connected graphs

Abstract

Let G be a connected graph of order n. The representation of a vertex v of G with respect to an ordered set W = {w1, w2, …, wk} is the k-vector r(v|W) = (d(v, w1), d(v, w2), …, d(v, wk)), where d(v, wi) represents the distance between vertices v and wi for 1 ≤ i ≤ k. An ordered set W is called a connected local resolving set of G if distinct adjacent vertices have distinct representations with respect to W, and the subgraph 〈W〉 induced by W is connected. A connected local resolving set of G of minimum cardinality is a connected local basis of G, and this cardinality is the connected local dimension cld(G) of G. Two vertices u and v of G are true twins if N[u] = N[v]. In this paper, we establish a fundamental property of a connected local basis of a connected graph G. We analyze the connected local dimension of a connected graph without a singleton true twin class and explore cases involving singleton true twin classes. Our investigation reveals that a graph of order n contains at most two non-singleton true twin classes when cld(G) = n − 2. Essentially, our work contributes to the characterization of graphs with a connected local dimension of n − 2.