Connected metric dimension of the class of ladder graphs

Batiha, M. Iqbal; Amin, Mohamed; Mohamed, Basma; Jebril, H. Iqbal

doi:10.21595/mme.2024.23934

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Math. models, eng.

2024

DOI: 10.21595/mme.2024.23934

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Connected metric dimension of the class of ladder graphs

M. Iqbal Batiha,

Mohamed Amin,

Basma Mohamed

et al.

Abstract: Numerous applications, like robot navigation, network verification and discovery, geographical routing protocols, and combinatorial optimization, make use of the metric dimension and connected metric dimension of graphs. In this work, the connected metric dimension types of ladder graphs, namely, ladder, circular, open, and triangular ladder graphs, as well as open diagonal and slanting ladder graphs, are studied.

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2024

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Cited by 2 publications

References 23 publications

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Secure metric dimension of new classes of graphs

Batiha,

Mohamed,

Jebril

2024

Math. models, eng.

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The metric representation of a vertex v of a graph G is a finite vector representing distances of v with respect to vertices of some ordered subset S⊆V (G). If no suitable subset of S provides separate representations for each vertex of V(G), then the set S is referred to as a minimal resolving set. The metric dimension of G is the cardinality of the smallest (with respect to its cardinality) minimal resolving set. A resolving set S is secure if for any v∈V–S, there exists x∈S such that (S–{x})∪{v} is a resolving set. For various classes of graphs, the value of the secure resolving number is determined and defined. The secure metric dimension of the graph classes is being studied in this work. The results show that different graph families have different metric dimensions.

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Secure metric dimension of new classes of graphs

Batiha,

Mohamed,

Jebril

2024

Math. models, eng.

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A special graph for the connected metric dimension of graphs

Batiha,

Anakira,

Hashim

et al. 2024

Math. models, eng.

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Given a connected graph G=(V, E), let d(x, y) represent the separation between x and y at its vertices. If each vertex in a collection B is uniquely identified by its vector of distances to the vertices in B, then that set of vertices resolves a graph G. A metric dimension of G is represented by dim(G) and is the smallest cardinality of a resolving set of G. If the subgraph B- induced by B is a nontrivial connected subgraph of G, then a resolving set B of G is connected. The metric dimension of G is the cardinality of the minimal resolving set, while the connected metric dimension of G is the cardinality of the smallest connected resolving set. The connected metric dimension of the knots graph, whitehead link graph and jewel graph are determined in this study. Finally, we derive the explicit formulas for the triangular book graph, quadrilateral book graph and crystal planar map.

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Connected metric dimension of the class of ladder graphs

Cited by 2 publications

References 23 publications

Secure metric dimension of new classes of graphs

Secure metric dimension of new classes of graphs

A special graph for the connected metric dimension of graphs

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