Ghebleh–Niepel conjecture on identifying codes in circulant graphs

Let $C_n(1,3)$ be the circulant graph on vertex set $\mathbb{Z}_n$ in which vertices at cyclic distances $1$ or $3$ are adjacent. An identifying code is a dominating vertex set whose closed neighborhoods intersected with the code are distinct for all vertices; let $\gamma^{ID}(C_n(1,3))$ denote the minimum size of such a code. Ghebleh–Niepel's conjecture. If $n$ is an integer such that $n \geq 19$ and $n \equiv 8 \pmod{11}$, then

$$\gamma^{ID}(C_n(1,3))=\lceil 4n/11\rceil+1.$$

The paper proves this conjecture and, more broadly, determines the exact identifying-code number for all $n\geq 11$, including additional residue classes for sufficiently large $n$.