Computational complexity aspects of super domination

Bujtás, Csilla (author), Ghanbari, Nima (author), Klavžar, Sandi (author)

Let $G$ be a graph. A dominating set $D\subseteq V(G)$ is a super dominating set if for every vertex $x\in V(G) \setminus D$ there exists $y\in D$ such that $N_G(y)\cap (V(G)\setminus D)) = \{x\}$. The cardinality of a smallest super dominating set of $G$ is the super domination number of $G$. An exact formula for the super domination number of a tree $T$ is obtained, and it is demonstrated that a smallest super dominating set of $T$ can be computed in linear time. It is proved that it is NP-complete to decide whether the super domination number of a graph $G$ is at most a given integer if $G$ is a bipartite graph of girth at least $8$. The super domination number is determined for all $k$-subdivisions of graphs. Interestingly, in half of the cases the exact value can be efficiently computed from the obtained formulas, while in the other cases the computation is hard. While obtaining these formulas, II-matching numbers are introduced and proved that they are computationally hard to determine.

• Language: English
• Keywords: super domination number, trees, bipartite graphs, k-subdivision of a graph, computational complexity, matching, II-matching number, II-matchings
• 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: 2023
• Number of pages: 14 str.
• Numbering: Vol. 975, art. 114137
• PID: 20.500.12556/RUL-153123
• UDC: 519.17
• ISSN on article: 0304-3975
• DOI: 10.1016/j.tcs.2023.114137
• COBISS.SI-ID: 162318851
• Publication date in RUL: 18.12.2023

Record is a part of a journal

• Title: Theoretical computer science
• Shortened title: Theor. comp. sci.
• Publisher: Elsevier
• ISSN: 0304-3975
• COBISS.SI-ID: 26525952

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: število super dominacije, drevesa, dvodelni grafi, k-subdivizija grafa, računska zahtevnost, prirejanje, število II-prirejanja