Total Weak Roman Domination in Graphs
Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain
CONACYT Research Fellow—Centro de Investigación en Matemáticas, 36023 Guanajuato, GTO, Mexico
Author to whom correspondence should be addressed.
Received: 17 May 2019 / Revised: 18 June 2019 / Accepted: 19 June 2019 / Published: 24 June 2019
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Given a graph
, a function
is said to be a total dominating function if
denotes the open neighbourhood of v
. We say that a function
is a total weak Roman dominating function if f
is a total dominating function and for every vertex
such that the function
, defined by
, is a total dominating function as well. The weight of a function f
is defined to be
In this article, we introduce the study of the total weak Roman domination number of a graph G
, denoted by
, 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
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.
This is an open access article distributed under the Creative Commons Attribution License
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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MDPI and ACS Style
Cabrera Martínez, A.; Montejano, L.P.; Rodríguez-Velázquez, J.A. Total Weak Roman Domination in Graphs. Symmetry 2019, 11, 831.
Cabrera Martínez A, Montejano LP, Rodríguez-Velázquez JA. Total Weak Roman Domination in Graphs. Symmetry. 2019; 11(6):831.
Cabrera Martínez, Abel; Montejano, Luis P.; Rodríguez-Velázquez, Juan A. 2019. "Total Weak Roman Domination in Graphs." Symmetry 11, no. 6: 831.
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