Author, as appears in the article.: Cabrera Martinez, A; Puertas, M L; Rodriguez-Velazquez, J A
Department: Enginyeria Informàtica i Matemàtiques
URV's Author/s: CABRERA MARTÍNEZ, ABEL / Rodríguez Velázquez, Juan Alberto
Keywords: Unique response roman domination Response roman domination Number Lexicographic product Efficient open domination Differential of a graph 2-packing differential
Abstract: 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.
Thematic Areas: Mathematics, applied Mathematics (miscellaneous) Mathematics Matemática / probabilidade e estatística Ensino Economia Applied mathematics
licence for use: https://creativecommons.org/licenses/by/3.0/es/
Author's mail: juanalberto.rodriguez@urv.cat
Author identifier: 0000-0002-9082-7647
Record's date: 2024-10-26
Papper version: info:eu-repo/semantics/publishedVersion
Link to the original source: https://link.springer.com/article/10.1007/s00025-021-01473-8
Licence document URL: https://repositori.urv.cat/ca/proteccio-de-dades/
Papper original source: Results In Mathematics. 76 (3): 157-
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
Article's DOI: 10.1007/s00025-021-01473-8
Entity: Universitat Rovira i Virgili
Journal publication year: 2021
Publication Type: Journal Publications