in the corresponding section. Finally, Section \[sec:conclusions\] concludes the article.

Preliminaries {#sec:prelim}
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In our article graphs are undirected, simple, and without loops. We use standard graph-theoretic notation; see for instance [@Die05]. We denote by $d_G(v,w)$ the distance in $G$ between two vertices $v$ and $w$ and by $d_G(W_1,W_2) = \min\{ d_G(w_1,w_2) : w_1\in W_1, w_2\in W_2 \}$ the distance between two sets of vertices $W_1$ and $W_2$ of $G$. Given $S \subseteq V(G)$, we denote by $N(S)$ the set of vertices in $V(G) \setminus S$ having at least one neighbor in $S$.

A parameterized graph problem ${\Pi}$ is called *packing-certifiable* if there exists a language $L^{\Pi}$ (called [*certifying language for $\Pi$*]{}) defined on pairs $(G,{\cal S})$, where $G$ is a graph and ${\cal S}$ is a collection of subgraphs of $G$, such that $(G,k)$ is a [<span style="font-variant:small-caps;">Yes</span>]{}-instance of $\Pi$ if and only if there exists a collection ${\cal S}$ of subgraphs of $G$ with $|{\cal S}| {\geqslant}k$ such that $(G,{\cal S}) \in L^{\Pi}$.

In the above definition, for the sake of generality we do not require the subgraphs in the collection ${\cal S}$ to be pairwise distinct. Also, note that the subclass of packing-certifiable problems where each subgraph 