The doubly metric dimensions of cactus graphs and block graphs

Kairui Nie () and Kexiang Xu ()
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Kairui Nie: Nanjing University of Aeronautics and Astronautics
Kexiang Xu: Computing of Air Vehicles

Journal of Combinatorial Optimization, 2024, vol. 47, issue 4, No 14, 17 pages

Abstract: 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.

Keywords: Doubly metric dimension; Doubly resolving set; Cactus graph; Block graph; 05C12 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10878-024-01168-0

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