| Title: | Revisiting $d$-distance (independent) domination in trees and in bipartite graphs |
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| Authors: | ID Bujtás, Csilla (Author)
ID Iršič Chenoweth, Vesna (Author)
ID Klavžar, Sandi (Author)
ID Zhang, Gang (Author) |
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| Files: |  PDF - Presentation file, download (856,17 KB)
MD5: 3378D59F4DD51C214FE3BE26D6F03864
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0012365X25005801
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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. |
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| Keywords: | d-distance dominating set, p-packing set, trees, bipartite graphs |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.06.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | 11 str. |
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| Numbering: | Vol. 349, iss. 6, article no. 114972 |
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| PID: | 20.500.12556/DiRROS-24967  |
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| UDC: | 519.17 |
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| ISSN on article: | 0012-365X |
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| DOI: | 10.1016/j.disc.2025.114972 |
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| COBISS.SI-ID: | 263511811 |
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| Note: |
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| Publication date in DiRROS: | 06.01.2026 |
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| Views: | 385 |
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| Downloads: | 187 |
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