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Total Weak Roman Domination in Graphs

  • Identification data

    Identifier: imarina:5745867
    Authors:
    Cabrera Martinez, AbelMontejano, Luis PRodriguez-Velazquez, Juan A
    Abstract:
    Given a graph G=(V,E) , a function f:V→{0,1,2,⋯} is said to be a total dominating function if ∑u∈N(v)f(u)>0 for every v∈V , where N(v) denotes the open neighbourhood of v. Let Vi={x∈V:f(x)=i} . We say that a function f:V→{0,1,2} is a total weak Roman dominating function if f is a total dominating function and for every vertex v∈V0 there exists u∈N(v)∩(V1∪V2) such that the function f′ , defined by f′(v)=1 , f′(u)=f(u)−1 and f′(x)=f(x) whenever x∈V\{u,v} , is a total dominating function as well. The weight of a function f is defined to be w(f)=∑v∈Vf(v). In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by γtr(G) , which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on γtr(G) and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard.
  • Others:

    Author, as appears in the article.: Cabrera Martinez, Abel; Montejano, Luis P; Rodriguez-Velazquez, Juan A
    Department: Enginyeria Informàtica i Matemàtiques
    e-ISSN: 2073-8994
    URV's Author/s: CABRERA MARTÍNEZ, ABEL / Montejano Cantoral, Luis Pedro / Rodríguez Velázquez, Juan Alberto
    Keywords: Weak roman domination Total roman domination Total domination Secure total domination Np-hard problem
    Abstract: Given a graph G=(V,E) , a function f:V→{0,1,2,⋯} is said to be a total dominating function if ∑u∈N(v)f(u)>0 for every v∈V , where N(v) denotes the open neighbourhood of v. Let Vi={x∈V:f(x)=i} . We say that a function f:V→{0,1,2} is a total weak Roman dominating function if f is a total dominating function and for every vertex v∈V0 there exists u∈N(v)∩(V1∪V2) such that the function f′ , defined by f′(v)=1 , f′(u)=f(u)−1 and f′(x)=f(x) whenever x∈V\{u,v} , is a total dominating function as well. The weight of a function f is defined to be w(f)=∑v∈Vf(v). In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by γtr(G) , which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on γtr(G) and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard.
    Thematic Areas: Visual arts and performing arts Physics and astronomy (miscellaneous) Multidisciplinary sciences Modeling and simulation Mathematics, interdisciplinary applications Mathematics (miscellaneous) Mathematics (all) Matemática / probabilidade e estatística General mathematics Engineering (miscellaneous) Computer science (miscellaneous) Ciência da computação Chemistry (miscellaneous) Arts and humanities (miscellaneous) Architecture Applied mathematics
    licence for use: thttps://creativecommons.org/licenses/by/3.0/es/
    ISSN: 20738994
    Author's mail: luispedro.montejano@urv.cat juanalberto.rodriguez@urv.cat
    Author identifier: 0000-0002-9082-7647
    Record's date: 2024-10-26
    Journal volume: 11
    Papper version: info:eu-repo/semantics/publishedVersion
    Link to the original source: https://www.mdpi.com/2073-8994/11/6/831
    Licence document URL: https://repositori.urv.cat/ca/proteccio-de-dades/
    Papper original source: Symmetry-Basel. 11 (6): 831-
    APA: Cabrera Martinez, Abel; Montejano, Luis P; Rodriguez-Velazquez, Juan A (2019). Total Weak Roman Domination in Graphs. Symmetry-Basel, 11(6), 831-. DOI: 10.3390/sym11060831
    Article's DOI: 10.3390/sym11060831
    Entity: Universitat Rovira i Virgili
    Journal publication year: 2019
    Publication Type: Journal Publications
  • Keywords:

    Applied Mathematics,Architecture,Arts and Humanities (Miscellaneous),Chemistry (Miscellaneous),Computer Science (Miscellaneous),Engineering (Miscellaneous),Mathematics (Miscellaneous),Mathematics, Interdisciplinary Applications,Modeling and Simulation,Multidisciplinary Sciences,Physics and Astronomy (Miscellaneous),Visual Arts and Performi
    Weak roman domination
    Total roman domination
    Total domination
    Secure total domination
    Np-hard problem
    Visual arts and performing arts
    Physics and astronomy (miscellaneous)
    Multidisciplinary sciences
    Modeling and simulation
    Mathematics, interdisciplinary applications
    Mathematics (miscellaneous)
    Mathematics (all)
    Matemática / probabilidade e estatística
    General mathematics
    Engineering (miscellaneous)
    Computer science (miscellaneous)
    Ciência da computação
    Chemistry (miscellaneous)
    Arts and humanities (miscellaneous)
    Architecture
    Applied mathematics
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