20.500.12556/DiRROS-27714 The $d$-distance $p$-packing domination number: complexity, cycles, and trees 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 $\gamma_2^2(T) \leq \gamma_2^2(G)$. d-distance dominating set p-packing set dominating set trees planar graphs d-razdaljno dominantna množica p-pakirna množica dominantna množica drevesa ravninski grafi true false true Angleški jezik Slovenski jezik Neznano 2026-02-23 09:38:38 2026-02-23 09:38:39 2026-03-09 09:04:10 0000-00-00 00:00:00 2026 0 0 22 str. iss. 2, article no. 25 Vol. 100 Apr. 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2026-04-01 519.17 0001-9054 10.1007/s00010-026-01266-w 269190659 s00010-026-01266-w.pdf s00010-026-01266-w.pdf 1 9F797C466E3B0F834BA95DCB376B70E7 6b746da0b599e441e143d6bf7de0f686d0ca1100e6393f2ec8d5ce72c8820d0b 514669c0-1093-11f1-9cb0-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=40710 https://link.springer.com/article/10.1007/s00010-026-01266-w 1 0fe55a91-1093-11f1-9cb0-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=40709 Inštitut za matematiko, fiziko in mehaniko 0 0 0