Research demonstrates the identifying code number of graph subdivisions; implications for graph theory are significant.
Let $G=(V, E)$ be a simple graph. A set C of vertices of G is an identifying code of G if for every two vertices x and y the sets NG[x] ∩ C and NG[y] ∩ C are non-empty and different. Given a graph $G,$ the smallest size of an identifying code of G is called the identifying code number of G and denoted by γID(G). In this paper, we prove that the identifying code number of the subdivision of a graph G of order n is at most n. Also, we prove that the identifying code number of the subdivision of graphs Kₙ, Kr,s and CP(s) are $n$, $r+s$ and $2s$, respectively. Finally, we conjecture that for every graph G of order n the identifying code number of the subdivision of G is n.
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Ahmadi et al. (2025) studied this question.
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