Partial domination in supercubic graphs

Bujtás, Csilla (author), Henning, Michael A. (author), Klavžar, Sandi (author)

For some $\alpha$ with $0 < \alpha \le 1$, a subset $X$ of vertices in a graph $G$ of order $n$ is an $\alpha$-partial dominating set of $G$ if the set $X$ dominates at least $\alpha \times n$ vertices in $G$. The $\alpha$-partial domination number ${\rm pd}_{\alpha}(G)$ of $G$ is the minimum cardinality of an $\alpha$-partial dominating set of $G$. In this paper partial domination of graphs with minimum degree at least $3$ is studied. It is proved that if $G$ is a graph of order $n$ and with $\delta(G)\ge 3$, then ${\rm pd}_{\frac{7}{8}}(G) \le \frac{1}{3}n$. If in addition $n\ge 60$, then ${\rm pd}_{\frac{9}{10}}(G) \le \frac{1}{3}n$, and if $G$ is a connected cubic graph of order $n\ge 28$, then ${\rm pd}_{\frac{13}{14}}(G) \le \frac{1}{3}n$. Along the way it is shown that there are exactly four connected cubic graphs of order $14$ with domination number $5$.

• Language: English
• Keywords: domination, partial domination, cubic graphs, supercubic graphs
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: 11 str.
• Numbering: Vol. 347, iss. 1, art. 113669
• PID: 20.500.12556/RUL-152915
• UDC: 519.17
• ISSN on article: 0012-365X
• DOI: 10.1016/j.disc.2023.113669
• COBISS.SI-ID: 162890243
• Publication date in RUL: 11.12.2023

Record is a part of a journal

• Title: Discrete mathematics
• Shortened title: Discrete math.
• Publisher: North-Holland
• ISSN: 0012-365X
• COBISS.SI-ID: 1118479

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.

Secondary language

• Language: Slovenian
• Keywords: dominacija, delna dominacija, kubični grafi, superkubični grafi