| Title: | The $d$-distance $p$-packing domination number: complexity, cycles, and trees |
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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 (634,30 KB)
MD5: 9F797C466E3B0F834BA95DCB376B70E7
 URL - Source URL, visit https://link.springer.com/article/10.1007/s00010-026-01266-w
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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: | A set of vertices $X\subseteq V(G)$ is a $d$-distance dominating set if for every $u\in V(G)\setminus X$ there exists $x\in X$ such that $d(u,x) \le d$, and $X$ is a $p$-packing if $d(u,v) \ge p+1$ for every different $u,v\in X$. 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. It is proved that for every two fixed integers $d$ and $p$ with $2 \le d$ and $0 \le p \leq 2d-1$, the decision problem whether $\gamma_d^p(G) \leq k$ holds is NP-complete for bipartite planar graphs. A necessary and sufficient condition for the existence of a $d$-distance $p$-packing dominating set in $C_n$ is obtained and $\gamma_d^p(C_n)$ determined for every $d$, $p$, and $n$. For a tree $T$ on $n$ vertices with $\ell$ leaves and $s$ support vertices it is proved that (i) $\gamma_2^0(T) \geq \frac{n-\ell-s+4}{5}$, (ii) $\left \lceil \frac{n-\ell-s+4}{5} \right \rceil \leq \gamma_2^2(T) \leq \left \lfloor \frac{n+3s-1}{5} \right \rfloor$, and if $d \geq 2$, then (iii) $\gamma_d^2(T) \leq \frac{n-2\sqrt{n}+d+1}{d}$. Inequality (i) improves an earlier bound due to Meierling and Volkmann, and independently Raczek, Lema\'nska, and Cyman, while (iii) extends an earlier result for $\gamma_2^2(T)$ due to Henning. Sharpness of the bounds is discussed and established in most cases. It is also proved that every connected graph $G$ contains a spanning tree $T$ such that $\gamma_2^2(T) \leq \gamma_2^2(G)$. |
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| Keywords: | d-distance dominating set, p-packing set, dominating set, trees, planar graphs |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.04.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | 22 str. |
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| Numbering: | Vol. 100, iss. 2, article no. 25 |
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| PID: | 20.500.12556/DiRROS-27714  |
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| UDC: | 519.17 |
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| ISSN on article: | 0001-9054 |
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| DOI: | 10.1007/s00010-026-01266-w |
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| COBISS.SI-ID: | 269190659 |
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| Publication date in DiRROS: | 23.02.2026 |
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| Views: | 179 |
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| Downloads: | 88 |
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