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Further Contributions on the Outer Multiset Dimension of Graphs

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Abstract

The outer multiset dimension \(\textrm{dim}_\textrm{ms}(G)\) of a graph G is the cardinality of a smallest set of vertices that uniquely recognize all the vertices outside this set by using multisets of distances to the set. It is proved that \(\textrm{dim}_\textrm{ms}(G) = n(G) - 1\) if and only if G is a regular graph with diameter at most 2. Graphs G with \(\textrm{dim}_\textrm{ms}(G)=2\) are described and recognized in polynomial time. A lower bound on the lexicographic product of G and H is proved when H is complete or edgeless, and the extremal graphs are determined. It is proved that \(\textrm{dim}_\textrm{ms}(P_s\,\square \, P_t) = 3\) for \(s\ge t\ge 2\).

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Funding

Sandi Klavžar acknowledges the financial support from the Slovenian Research Agency through research core funding No. P1-0297 and projects J1-2452 and N1-0285. Dorota Kuziak and Ismael G. Yero have been partially supported by the Spanish Ministry of Science and Innovation through the grant PID2019-105824GB-I00. Moreover, this work was initiated while the first author Sandi Klavžar was visiting the University of Cadiz with the support of the grant PID2019-105824GB-I00. Dorota Kuziak has also been partially supported by “Plan Propio de Investigación-UCA” ref. no. EST2022-075.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Sandi Klavžar

  2. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia

    Sandi Klavžar

  3. Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Sandi Klavžar

  4. Departamento de Estadística e Investigación Operativa, Universidad de Cádiz, Algeciras, Spain

    Dorota Kuziak

  5. Departamento de Matemáticas, Universidad de Cádiz, Algeciras, Spain

    Ismael G. Yero

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  1. Sandi Klavžar
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  2. Dorota Kuziak
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  3. Ismael G. Yero
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Correspondence to Sandi Klavžar.

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Klavžar, S., Kuziak, D. & Yero, I.G. Further Contributions on the Outer Multiset Dimension of Graphs. Results Math 78, 50 (2023). https://doi.org/10.1007/s00025-022-01829-8

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