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On the 2-Packing Differential of a Graph

  • Dades identificatives

    Identificador: imarina:9227041
    Autors:
    Cabrera Martinez, APuertas, M LRodriguez-Velazquez, J A
    Resum:
    Let G be a graph of order n(G) and vertex set V(G). Given a set S subset of V (G), we define the external neighbourhood of S as the set Ne(S) of all vertices in V (G)\S having at least one neighbour in S. The differential of S is defined to be partial derivative(S) = vertical bar N-e(S)vertical bar - vertical bar S vertical bar. In this paper, we introduce the study of the 2-packing differential of a graph, which we define as partial derivative(2p)(G) = max{partial derivative(S) : S subset of V(G) is a 2-packing}. We show that the 2-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of 2-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions. Among other results, we obtain a Gallai-type theorem, which states that partial derivative(2p)(G)+ mu(R)(G) = n(G), where mu(R)(G) denotes the unique response Roman domination number of G. As a consequence of the study, we derive several combinatorial results on mu(R)(G), and we show that the problem of finding this parameter is NP-hard. In addition, the particular case of lexicographic product graphs is discussed.
  • Altres:

    Autor segons l'article: Cabrera Martinez, A; Puertas, M L; Rodriguez-Velazquez, J A
    Departament: Enginyeria Informàtica i Matemàtiques
    Autor/s de la URV: CABRERA MARTÍNEZ, ABEL / Rodríguez Velázquez, Juan Alberto
    Paraules clau: Unique response roman domination Response roman domination Number Lexicographic product Efficient open domination Differential of a graph 2-packing differential
    Resum: Let G be a graph of order n(G) and vertex set V(G). Given a set S subset of V (G), we define the external neighbourhood of S as the set Ne(S) of all vertices in V (G)\S having at least one neighbour in S. The differential of S is defined to be partial derivative(S) = vertical bar N-e(S)vertical bar - vertical bar S vertical bar. In this paper, we introduce the study of the 2-packing differential of a graph, which we define as partial derivative(2p)(G) = max{partial derivative(S) : S subset of V(G) is a 2-packing}. We show that the 2-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of 2-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions. Among other results, we obtain a Gallai-type theorem, which states that partial derivative(2p)(G)+ mu(R)(G) = n(G), where mu(R)(G) denotes the unique response Roman domination number of G. As a consequence of the study, we derive several combinatorial results on mu(R)(G), and we show that the problem of finding this parameter is NP-hard. In addition, the particular case of lexicographic product graphs is discussed.
    Àrees temàtiques: Mathematics, applied Mathematics (miscellaneous) Mathematics Matemática / probabilidade e estatística Ensino Economia Applied mathematics
    Accès a la llicència d'ús: https://creativecommons.org/licenses/by/3.0/es/
    Adreça de correu electrònic de l'autor: juanalberto.rodriguez@urv.cat
    Identificador de l'autor: 0000-0002-9082-7647
    Data d'alta del registre: 2024-10-26
    Versió de l'article dipositat: info:eu-repo/semantics/publishedVersion
    Enllaç font original: https://link.springer.com/article/10.1007/s00025-021-01473-8
    URL Document de llicència: https://repositori.urv.cat/ca/proteccio-de-dades/
    Referència a l'article segons font original: Results In Mathematics. 76 (3): 157-
    Referència de l'ítem segons les normes APA: Cabrera Martinez, A; Puertas, M L; Rodriguez-Velazquez, J A (2021). On the 2-Packing Differential of a Graph. Results In Mathematics, 76(3), 157-. DOI: 10.1007/s00025-021-01473-8
    DOI de l'article: 10.1007/s00025-021-01473-8
    Entitat: Universitat Rovira i Virgili
    Any de publicació de la revista: 2021
    Tipus de publicació: Journal Publications
  • Paraules clau:

    Applied Mathematics,Mathematics,Mathematics (Miscellaneous),Mathematics, Applied
    Unique response roman domination
    Response roman domination
    Number
    Lexicographic product
    Efficient open domination
    Differential of a graph
    2-packing differential
    Mathematics, applied
    Mathematics (miscellaneous)
    Mathematics
    Matemática / probabilidade e estatística
    Ensino
    Economia
    Applied mathematics
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