11 problems

Let $G$ be a twin-free graph, let $|V(G)|$ denote its number of vertices, and let $amma^{\text{\tiny{ID}}}(G)$ be its identifying code number. Write $\Delta(G)$ for the maximum deg…

Let $K_m^3=K_m\square K_m\square K_m$ be the 3-dimensional $m$-ary Hamming cube, and let $\gamma^{ID}(G)$ denote the minimum size of an identifying code in a graph $G$. Goddard–Was…

Ghebleh–Niepel conjecture. In the cases not covered by their exact-value results, the lower bound in each inequality should be increased by one, so that the corresponding upper-bou…

Let $G$ be a connected closed-twin-free graph with $n$ vertices and maximum degree $\Delta$. An identifying code of $G$ is a set of vertices whose intersections with the closed nei…

Let $G$ be a connected identifiable graph of order $n\ge 2$ and maximum degree $\Delta\ge 2$. Write $\gamma^{{\rm ID}}(G)$ for the minimum size of an identifying code of $G$. Ident…

Let $\mathbb{F}^n$ and $\mathbb{F}^m$ be binary Hamming spaces, and let $Id_r(\mathbb{F}^n)$ denote the minimum cardinality of an identifying code of radius $r$ in $\mathbb{F}^n$.…

Let $G$ be a cubic graph, and let $\textrm{DET:IC\%}(G)$ denote the minimum density of a deterministic error-detecting identifying code in $G$. Lower-bound conjecture. … The paper…

Twin-free graph bounds conjecture. The two bounds

Let $C_n(1,d)$ be the circulant graph on vertex set $\mathbb{Z}_n$ in which vertices at cyclic distances $1$ or $d$ are adjacent, with $d>3$. Identifying and locating-dominating co…

Let $G$ be a connected cubic graph admitting an identifying open code, meaning an identifying code defined using open balls, so vertices do not dominate or identify themselves. Let…

Identifying-code extremal conjecture.