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 \(0 \le p \le 2d-1\) , the decision problem whether \(\gamma _d^p(G) \le k\) 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) \(\gamma _2^0(T) \ge \frac{n-\ell -s+4}{5}\) , (ii) \(\lceil \frac{n-\ell -s+4}{5} \rceil \le \gamma _2^2(T) \le \left\lfloor \frac{n+3s-1}{5} \right\rfloor \) , and if \(d \ge 2\) , then (iii) \(\gamma _d^2(T) \le \frac{n-2\sqrt{n}+d+1}{d}\) . Inequality (i) improves an earlier bound due to Meierling and Volkmann, and independently Raczek, Lemańska, 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 \(\gamma _2^2(T) \le \gamma _2^2(G)\) .
