The effect of vertex and edge deletion on the edge metric dimension of graphs

Meiqin Wei (), Jun Yue () and Lily Chen ()
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Meiqin Wei: Shanghai Maritime University
Jun Yue: Tiangong University
Lily Chen: Huaqiao University

Journal of Combinatorial Optimization, 2022, vol. 44, issue 1, No 16, 342 pages

Abstract: Abstract Let $$G=(V(G),E(G))$$ G = ( V ( G ) , E ( G ) ) be a connected graph. A set of vertices $$S\subseteq V(G)$$ S ⊆ V ( G ) is an edge metric generator of G if any pair of edges in G can be distinguished by their distance to a vertex in S. The edge metric dimension edim(G) is the minimum cardinality of an edge metric generator of G. In this paper, we first give a sharp bound on $$edim(G-e)-edim(G)$$ e d i m ( G - e ) - e d i m ( G ) for a connected graph G and any edge $$e\in E(G)$$ e ∈ E ( G ) . On the other hand, we show that the value of $$edim(G)-edim(G-e)$$ e d i m ( G ) - e d i m ( G - e ) is unbounded for some graph G and some edge $$e\in E(G)$$ e ∈ E ( G ) . However, for a unicyclic graph H, we obtain that $$edim(H)-edim(H-e)\le 1$$ e d i m ( H ) - e d i m ( H - e ) ≤ 1 , where e is an edge of the unique cycle in H. And this conclusion generalizes the result on the edge metric dimension of unicyclic graphs given by Knor et al. Finally, we construct graphs G and H such that both $$edim(G)-edim(G-u)$$ e d i m ( G ) - e d i m ( G - u ) and $$edim(H-v)-edim(H)$$ e d i m ( H - v ) - e d i m ( H ) can be arbitrarily large, where $$u\in V(G)$$ u ∈ V ( G ) and $$v\in V(H)$$ v ∈ V ( H ) .

Keywords: Edge metric generator; Vertex deletion; Edge deletion; Edge metric dimension; 05C69; 05C07 (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10878-021-00838-7

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