Computing Metric and Connected Metric Dimension of Some Graphs

For a connected graph G=(V, E), a set of vertices B ⊆ V(G) resolves G if every vertex of G is uniquely determined by its vector of distances to the vertices in B. Mathematically: r(v | B) = (d(v, x₁), d(v, x₂), ..., d(v, xₖ)) is unique for every v in V(G). A metric basis B of G is connected if the s...

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Published inEuropean journal of pure and applied mathematics Vol. 18; no. 3; p. 6438
Main Authors Elrokh, Ashraf, S. Almotairi, Eman, Mostafa, Hoda
Format Journal Article
LanguageEnglish
Published 01.08.2025
Online AccessGet full text
ISSN1307-5543
1307-5543
DOI10.29020/nybg.ejpam.v18i3.6438

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Abstract For a connected graph G=(V, E), a set of vertices B ⊆ V(G) resolves G if every vertex of G is uniquely determined by its vector of distances to the vertices in B. Mathematically: r(v | B) = (d(v, x₁), d(v, x₂), ..., d(v, xₖ)) is unique for every v in V(G). A metric basis B of G is connected if the subgraph induced by B is a nontrivial connected subgraph of G. The cardinality number of the connected metric basis is the connected metric dimension of G and is denoted cdim(G). We will introduce connected metric some graphs and proof them indexing, abstracting, and retrieval purposes. Also we proposing an approximate algorithm which finds a minimum connected metric dimension of a given graph.
AbstractList For a connected graph G=(V, E), a set of vertices B ⊆ V(G) resolves G if every vertex of G is uniquely determined by its vector of distances to the vertices in B. Mathematically: r(v | B) = (d(v, x₁), d(v, x₂), ..., d(v, xₖ)) is unique for every v in V(G). A metric basis B of G is connected if the subgraph induced by B is a nontrivial connected subgraph of G. The cardinality number of the connected metric basis is the connected metric dimension of G and is denoted cdim(G). We will introduce connected metric some graphs and proof them indexing, abstracting, and retrieval purposes. Also we proposing an approximate algorithm which finds a minimum connected metric dimension of a given graph.
Author Elrokh, Ashraf
Mostafa, Hoda
S. Almotairi, Eman
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