in ${\cal S}$ is restricted to consist of a single vertex corresponds to the class of vertex-certifiable problems defined in [@KviaDP].

For a class of graphs [${\mathcal}{G}$]{}, we denote by $\Pi_{\ensuremath{{\mathcal}{G}}\xspace}$ the problem $\Pi$ where the instances are restricted to contain graphs belonging to [${\mathcal}{G}$]{}. With a packing-certifiable problem we can associate in a natural way an optimization function as follows.

\[defi:optimization-function\] Given a packing-certifiable parameterized problem $\Pi$, the *maximization function* $f^{\Pi}: \Gamma^* \rightarrow \mathbb{N}\cup\{-\infty\}$ is defined as $$\begin{aligned}
f^{\Pi}(G)=\
\left\{\begin{array}{lll}
  & \max\{|{\cal S}| : (G,{\cal S}) \in L^{\Pi}\} & \mbox{, if there exists such an ${\cal S}$ and}\\
  & - \infty                     & \mbox{, otherwise}.
\end{array}\right.\end{aligned}$$

\[defi:boundaried\] A *boundaried graph* is a graph $G$ with a set $B \subseteq V (G)$ of distinguished vertices and an injective labeling $\lambda_G: B \to \mathbb{N}$. The set $B$ is called the *boundary* of $G$ and it is denoted by $\partial(G)$. The set of labels is denoted by $\Lambda(G) = \{\lambda_G(v) : v \in \partial(G) \}$. We say that a boundaried graph is a *$t$-boundaried graph* if $\Lambda(G) \subseteq \{1, \ldots ,t\}$.

We denote by [${\mathcal}{B}_t$]{}the set of all $t$-boundaried graphs.

\[defi:gluing\] Let $G_1$ and $G_2$ be two boundaried graphs. We denote by $G_1 