Revisiting $d$-distance (independent) domination in trees and in bipartite graphs

Bujtás, Csilla (author), Iršič Chenoweth, Vesna (author), Klavžar, Sandi (author), Zhang, Gang (author)

The $d$-distance $p$-packing domination number $\gamma_d^p(G)$ of $G$ is the minimum size of a set of vertices of $G$ which is both a $d$-distance dominating set and a $p$-packing. In 1994, Beineke and Henning conjectured that if $d\ge 1$ and $T$ is a tree of order $n \geq d+1$, then $\gamma_d^1(T) \leq \frac{n}{d+1}$. They supported the conjecture by proving it for $d\in \{1,2,3\}$. In this paper, it is proved that $\gamma_d^1(G) \leq \frac{n}{d+1}$ holds for any bipartite graph $G$ of order $n \geq d+1$, and any $d\ge 1$. Trees $T$ for which $\gamma_d^1(T) = \frac{n}{d+1}$ holds are characterized. It is also proved that if $T$ has $\ell$ leaves, then $\gamma_d^1(T) \leq \frac{n-\ell}{d}$ (provided that $n-\ell \geq d$), and $\gamma_d^1(T) \leq \frac{n+\ell}{d+2}$ (provided that $n\geq d$). The latter result extends Favaron's theorem from 1992 asserting that $\gamma_1^1(T) \leq \frac{n+\ell}{3}$. In both cases, trees that attain the equality are characterized and relevant conclusions for the $d$-distance domination number of trees derived.

• Language: English
• Keywords: d-distance dominating set, p-packing set, trees, bipartite 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
• Publication date: 01.06.2026
• Year: 2026
• Number of pages: 11 str.
• Numbering: Vol. 349, iss. 6, art. 114972
• PID: 20.500.12556/RUL-177749
• UDC: 519.17
• ISSN on article: 0012-365X
• DOI: 10.1016/j.disc.2025.114972
• COBISS.SI-ID: 263511811
• Publication date in RUL: 06.01.2026

Record is a part of a journal

• Title: Discrete mathematics
• Shortened title: Discrete math.
• Publisher: Elsevier
• 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: d-razdaljna dominantna množica, p-pakirna množica, drevesa, dvodelni grafi