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New Optimal Results on Codes for Location in Graphs

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arxiv 2306.07862 v2 pith:5NUWSYTW submitted 2023-06-13 cs.DM math.CO

Ville Junnila , Tero Laihonen , Tuomo Lehtilä This is my paper · ORCID
classification cs.DMmath.CO
keywords codesgraphsoptimalsolid-locating-dominatingself-locating-dominatingsquarebroadencomplete
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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 $q\geq2$.

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  1. Metric Dimension of a Direct Product of Three Complete Graphs: The Middle Cone Family

    math.CO 2025-07 conditional novelty 6.0 of 10

    For direct products of three complete graphs whose dimensions lie in the middle cone, the metric dimension and location-total-domination number of K(n+1) both equal 2(n3+1)-1.

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