@article{Isariyapalakul2024, 
author = {Supachoke Isariyapalakul and Witsarut Pho-on and Varanoot Khemmani},
title = {The true twin classes-based investigation for connected local dimensions of connected graphs},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {4},
pages = {9435-9446},
keywords = {representation, connected local resolving set, connected local dimension, true twin graph},
url = {https://www.sciopen.com/article/10.3934/math.2024460},
doi = {10.3934/math.2024460},
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  =  {      w    1    ,      w    2    ,  .  .  .  ,      w    k    } is the    k-vector    r  (  v      |    W  )  =  (  d  (  v  ,      w    1    )  ,  d  (  v  ,      w    2    )  ,  .  .  .  ,  d  (  v  ,      w    k    )  ), where    d  (  v  ,      w    i    ) represents the distance between vertices    v and        w    i   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.}
}