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The doubly metric dimensions of cactus graphs and block graphs

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Abstract

Given a connected graph G, two vertices \(u,v\in V(G)\) doubly resolve \(x,y\in V(G)\) if \(d_{G}(x,u)-d_{G}(y,u)\ne d_{G}(x,v)-d_{G}(y,v)\). The doubly metric dimension \(\psi (G)\) of G is the cardinality of a minimum set of vertices that doubly resolves each pair of vertices from V(G). It is well known that deciding the doubly metric dimension of G is NP-hard. In this work we determine the exact values of doubly metric dimensions of unicyclic graphs which completes the known result. Furthermore, we give formulae for doubly metric dimensions of cactus graphs and block graphs.

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Acknowledgements

We sincerely thank two anonymous referees for their careful reading and some helpful comments on our paper.

Funding

This work was supported by NNSF of China (Grant No. 12271251).

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Authors and Affiliations

  1. School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 210016, People’s Republic of China

    Kairui Nie

  2. MIIT Key Laboratory of Mathematical Modelling and High Performance, Computing of Air Vehicles, Nanjing, 210016, People’s Republic of China

    Kexiang Xu

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  1. Kairui Nie
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  2. Kexiang Xu
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Correspondence to Kexiang Xu.

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Nie, K., Xu, K. The doubly metric dimensions of cactus graphs and block graphs. J Comb Optim 47, 67 (2024). https://doi.org/10.1007/s10878-024-01168-0

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