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The doubly metric dimensions of cactus graphs and block graphs

Author

Listed:
  • Kairui Nie

    (Nanjing University of Aeronautics and Astronautics)

  • Kexiang Xu

    (Computing of Air Vehicles)

Abstract

Given a connected graph G, two vertices $$u,v\in V(G)$$ u , v ∈ V ( G ) doubly resolve $$x,y\in V(G)$$ x , y ∈ V ( G ) if $$d_{G}(x,u)-d_{G}(y,u)\ne d_{G}(x,v)-d_{G}(y,v)$$ d G ( x , u ) - d G ( y , u ) ≠ d G ( x , v ) - d G ( y , v ) . The doubly metric dimension $$\psi (G)$$ ψ ( 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.

Suggested Citation

  • Kairui Nie & Kexiang Xu, 2024. "The doubly metric dimensions of cactus graphs and block graphs," Journal of Combinatorial Optimization, Springer, vol. 47(4), pages 1-17, May.
    • Handle: RePEc:spr:jcomop:v:47:y:2024:i:4:d:10.1007_s10878-024-01168-0
    DOI: 10.1007/s10878-024-01168-0
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