On the double Roman domination in generalized Petersen graphs P(5k, k)

Rupnik Poklukar, Darja (author), Žerovnik, Janez (author)

A double Roman dominating function on a graph G = (V, E) is a function f : V → {0, 1, 2, 3} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and every vertex u with f(u) = 1 is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f) = ∑$_{v∈V}$ f(v). The double Roman domination number γ$_{dR}$(G) of a graph G equals the minimum weight of a double Roman dominating function of G. We obtain closed expressions for the double Roman domination number of generalized Petersen graphs P(5k, k). It is proven that γ$_{dR}$(P(5k, k)) = 8k for k ≡ 2, 3 mod 5 and 8k ≤ γ$_{dR}$(P(5k, k)) ≤ 8k + 2 for k ≡ 0, 1, 4 mod 5. We also improve the upper bounds for generalized Petersen graphs P(20k, k).

• Language: English
• Keywords: double Roman domination, generalized Petersen graph, discharging method, graph cover, double Roman graph
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FS - Faculty of Mechanical Engineering
• Publication status: Published
• Publication version: Version of Record
• Year: 2022
• Number of pages: 19 str.
• Numbering: Vol. 10, iss. 1, art. 119
• PID: 20.500.12556/RUL-134466
• UDC: 519.17
• ISSN on article: 2227-7390
• DOI: 10.3390/math10010119
• COBISS.SI-ID: 93020931
• Publication date in RUL: 17.01.2022

Record is a part of a journal

• Title: Mathematics
• Shortened title: Mathematics
• Publisher: MDPI AG
• ISSN: 2227-7390
• COBISS.SI-ID: 523267865

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.
• Licensing start date: 01.01.2022

Secondary language

• Language: Slovenian
• Keywords: dvojna rimska dominacija, posplošeni Petersonovi grafi, pokritja grafov