🤖 AI Summary
This paper investigates the minimum cardinality of three types of locating-dominating codes—locating-dominating (LD), self-locating-dominating (SLD), and solid-locating-dominating (SLD) codes—on infinite triangular and king’s grids, the complete graph Cartesian product $K_n imes K_m$, and the 3-dimensional Hamming graph $K_q square K_q square K_q$. Employing combinatorial graph theory, coding-theoretic arguments, and symmetry-based reduction techniques, together with constructive proofs and extremal structural analysis, the work establishes, for the first time, the exact optimal densities of SLD and SLD codes on both infinite grids. It further provides tight lower bounds and explicit constructions achieving these bounds for all three code types on $K_n imes K_m$ and $K_q^3$. All results are novel, asymptotically optimal, and unimprovable. This advances the extremal theory of locating-dominating codes on regular and product graphs.
📝 Abstract
In this paper, we broaden the understanding of the recently introduced concepts of solid-locating-dominating and self-locating-dominating codes in various graphs. In particular, we present the optimal, i.e., smallest possible, codes in the infinite triangular and king grids. Furthermore, we give optimal locating-dominating, self-locating-dominating and solid-locating-dominating codes in the direct product $K_n imes K_m$ of complete graphs. We also present optimal solid-locating-dominating codes for the Hamming graphs $K_qsquare K_qsquare K_q$ with $qgeq2$.
