(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 an $(\alpha k,t)$-protrusion decomposition of $G$, we can construct a linear kernel for $\Pi$ of size at most $ (1+b({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace}))\cdot \alpha \cdot k$, where $b({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace})$ is the function defined in Lemma \[lem:progres size\].

Main ideas for the applications {#sec:applications}
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In this section by sketch the main ingredients that we use in our applications for obtaining the linear kernels, before going through the details for each problem in the next sections.

**<span style="font-variant:small-caps;">General methodology</span>.** The next theorem will be fundamental in the applications.

\[theo:prot dec\] Let $c,t$ be two positive integers, let $H$ be an $h$-vertex graph, let $G$ be an $n$-vertex $H$-topological-minor-free graph, and let $k$ be a positive integer. If we are given a set $X \subseteq V(G)$ with $|X| {\leqslant}c \cdot k$ such that ${{\mathbf{tw}}}(G-X) {\leqslant}t$, then we can compute in time $O(n)$ an $((\alpha_{H} \cdot t \cdot c)\cdot k, 2t + h)$-protrusion decomposition of $G$, where $\alpha_{H}$ is a constant depending only on 