such that $Y {\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace} Y'$ and ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}(Y',Y) {\leqslant}0$. Furthermore, such a protrusion can be computed in time $ O(|Y|)$, where the hidden constant depends only on ${{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,b,{\ensuremath{{\mathcal}{G}}\xspace}$, and $t$.

Let us now piece everything together to state the main result of [@KviaDP] that we need to reprove here for packing-certifiable problems. For issues of constructibility, we restrict [${\mathcal}{G}$]{}to be the class of $H$-(topological)-minor-free graphs.

$[\star]$ \[theo:main\] Let [${\mathcal}{G}$]{}be the class of graphs excluding some fixed graph $H$ as a (topological) minor and let $\Pi$ be a parameterized packing-certifiable problem defined on [${\mathcal}{G}$]{}. Let [[${\mathcal}{E}_{}$]{}]{}be an encoder, let $g: \mathbb{N} \rightarrow \mathbb{N}$, and let $t \in \mathbb{N}$ such that [[${\mathcal}{E}_{}$]{}]{}is a $g$-confined $\Pi$-encoder and [[${\mathcal}{E}_{}$]{}]{}is DP-friendly. Given an instance $(G,k)$ of $\Pi$ and a $t$-protrusion $Y$ in $G$, we can compute in time $O(|Y|)$ an equivalent instance $(G-(Y-\partial(Y)) \oplus Y',k')$ where $Y'$ is a $t$-protrusion with $|Y'| {\leqslant}b({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace})$ and $k'{\leqslant}k$ and where $b({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace})$ is the function defined in Lemma \[lem:progres size\].

Such a protrusion replacer can be used to obtain a kernel when, for instance, one is able to provide a protrusion decomposition of the instance.

$[\star]$ \[coro:main\] Let [${\mathcal}{G}$]{}be the class of graphs excluding some fixed graph $H$ as a 