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WEAK TOTAL RESOLVABILITY IN GRAPHS

  • Identification data

    Identifier: imarina:9295799
    Authors:
    Casel, KatrinEstrada-Moreno, AlejandroFernau, HenningAlberto Rodriguez-Velazquez, Juan
    Abstract:
    A vertex v is an element of V(G) is said to distinguish two vertices x, y is an element of V(G) of a graph G if the distance from v to x is different from the distance from v to y. A set W subset of V(G) is a total resolving set for a graph G if for every pair of vertices x, y is an element of V(G), there exists some vertex w is an element of W {x, y} which distinguishes x and y, while W is a weak total resolving set if for every x is an element of V(G) W and y is an element of W, there exists some w is an element of W {y} which distinguishes x and y. A weak total resolving set of minimum cardinality is called a weak total metric basis of G and its cardinality the weak total metric dimension of G. Our main contributions are the following ones: (a) Graphs with small and large weak total metric bases are characterised. (b) We explore the (tight) relation to independent 2-domination. (c) We introduce a new graph parameter, called weak total adjacency dimension and present results that are analogous to those presented for weak total dimension. (d) For trees, we derive a characterisation of the weak total (adjacency) metric dimension. Also, exact figures for our parameters are presented for (generalised) fans and wheels. (e) We show that for Cartesian product graphs, the weak total (adjacency) metric dimension is usually pretty small. (f) The weak total (adjacency) dimension is studied for lexicographic products of graphs.
  • Others:

    Author, as appears in the article.: Casel, Katrin; Estrada-Moreno, Alejandro; Fernau, Henning; Alberto Rodriguez-Velazquez, Juan
    Department: Enginyeria Informàtica i Matemàtiques
    URV's Author/s: Estrada Moreno, Alejandro / Rodríguez Velázquez, Juan Alberto
    Keywords: Weak total resolving set Weak total metric dimension Resolving set Metric dimension Graph operations Adjacency dimension
    Abstract: A vertex v is an element of V(G) is said to distinguish two vertices x, y is an element of V(G) of a graph G if the distance from v to x is different from the distance from v to y. A set W subset of V(G) is a total resolving set for a graph G if for every pair of vertices x, y is an element of V(G), there exists some vertex w is an element of W {x, y} which distinguishes x and y, while W is a weak total resolving set if for every x is an element of V(G) W and y is an element of W, there exists some w is an element of W {y} which distinguishes x and y. A weak total resolving set of minimum cardinality is called a weak total metric basis of G and its cardinality the weak total metric dimension of G. Our main contributions are the following ones: (a) Graphs with small and large weak total metric bases are characterised. (b) We explore the (tight) relation to independent 2-domination. (c) We introduce a new graph parameter, called weak total adjacency dimension and present results that are analogous to those presented for weak total dimension. (d) For trees, we derive a characterisation of the weak total (adjacency) metric dimension. Also, exact figures for our parameters are presented for (generalised) fans and wheels. (e) We show that for Cartesian product graphs, the weak total (adjacency) metric dimension is usually pretty small. (f) The weak total (adjacency) dimension is studied for lexicographic products of graphs.
    Thematic Areas: Mathematics Matemática / probabilidade e estatística Discrete mathematics and combinatorics Ciência da computação Applied mathematics
    licence for use: https://creativecommons.org/licenses/by/3.0/es/
    Author's mail: alejandro.estrada@urv.cat juanalberto.rodriguez@urv.cat
    Author identifier: 0000-0001-9767-2177 0000-0002-9082-7647
    Record's date: 2024-10-26
    Papper version: info:eu-repo/semantics/publishedVersion
    Link to the original source: https://www.dmgt.uz.zgora.pl/publish/volume.php?volume=36_1
    Licence document URL: https://repositori.urv.cat/ca/proteccio-de-dades/
    Papper original source: Discussiones Mathematicae Graph Theory. 36 (1): 185-210
    APA: Casel, Katrin; Estrada-Moreno, Alejandro; Fernau, Henning; Alberto Rodriguez-Velazquez, Juan (2016). WEAK TOTAL RESOLVABILITY IN GRAPHS. Discussiones Mathematicae Graph Theory, 36(1), 185-210. DOI: 10.7151/dmgt.1853
    Article's DOI: 10.7151/dmgt.1853
    Entity: Universitat Rovira i Virgili
    Journal publication year: 2016
    Publication Type: Journal Publications
  • Keywords:

    Applied Mathematics,Discrete Mathematics and Combinatorics,Mathematics
    Weak total resolving set
    Weak total metric dimension
    Resolving set
    Metric dimension
    Graph operations
    Adjacency dimension
    Mathematics
    Matemática / probabilidade e estatística
    Discrete mathematics and combinatorics
    Ciência da computação
    Applied mathematics
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