New Optimal Results on Codes for Location in Graphs
Ville Junnila, Tero Laihonen, Tuomo Lehtil\"a
TL;DR
This paper determines the smallest possible solid-locating-dominating and related codes in various graph classes, including infinite grids and product graphs, advancing the understanding of locating codes in graph theory.
Contribution
It provides the first known optimal codes for solid-locating-dominating and related concepts in several complex graph structures.
Findings
Optimal codes in infinite triangular and king grids.
Optimal locating-dominating and self-locating-dominating codes in $K_n\times K_m$.
Optimal solid-locating-dominating codes in Hamming graphs $K_q\square K_q\square K_q$.
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 of complete graphs. We also present optimal solid-locating-dominating codes for the Hamming graphs with .
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Taxonomy
TopicsCoding theory and cryptography · Cooperative Communication and Network Coding