New Optimal Results on Codes for Location in Graphs

TL;DR

The article advances the theory of locating-dominating codes by delivering tight optimality results across several graph families. It determines exact optimal densities and constructions for self- and solid-locating-dominating codes on the infinite triangular grid ($D=\tfrac{1}{4}$) and the infinite king grid ($D=\tfrac{1}{3}$), including novel global arguments for the lower bounds. It provides exact cardinalities for locating-dominating, self-locating-dominating, and solid-locating-dominating codes in the direct product $K_n\times K_m$ (with connections to the Cartesian product) and shows that $\gamma^{DLD}(K_n\times K_m)=\gamma^{DLD}(K_n\square K_m)$ and $\gamma^{SLD}(K_n\times K_m)=\gamma^{SLD}(K_n\square K_m)$ with explicit case analyses. Finally, it establishes that $\gamma^{DLD}(K^3_q)=q^2$ for all $q\ge 2$, using a layer-and-pipe framework to rule out smaller constructions. These results collectively enhance understanding of optimal location-dominating schemes in both infinite grids and product graphs, with potential implications for sensor-network monitoring strategies.

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 $K_n\times K_m$ of complete graphs. We also present optimal solid-locating-dominating codes for the Hamming graphs $K_q\square K_q\square K_q$ with $q\geq2$.

Figures (3)

• Figure 1: Triangular grid with the vertices $v = v(0,0)$, $u = v(1,-1)$ and $w = v(1,1)$.
• Figure 2: The darkened squares form a solid-locating-dominating code of density $\frac{1}{3}$ in the king grid.
• Figure 7: Optimal locating-dominating code for $K_{10}\times K_{10}$. Dark boxes are codewords.

Theorems & Definitions (23)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4: JLLrntcld
• Corollary 1.5
• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.4
• Lemma 2.5