an encoder [[${\mathcal}{E}_{}$]{}]{}and an equivalence class $\mathfrak{C} \subseteq {\ensuremath{{\mathcal}{F}_t}\xspace}$ of [[${\mathcal}{E}_{}$]{}]{}, a graph $G \in \mathfrak{C}$ is a *progressive representative* of $\mathfrak{C}$ if for any $G'\in \mathfrak{C}$, it holds that ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}(G,G') {\leqslant}0$.

$[\star]$\[lem:progres size\] Let [${\mathcal}{G}$]{}be a class of graphs whose membership is expressible in MSO logic. For any encoder [[${\mathcal}{E}_{}$]{}]{}, any function $g: \mathbb{N} \rightarrow \mathbb{N}$, and any $t \in \mathbb{N}$, if [[${\mathcal}{E}_{}$]{}]{}is $g$-confined and [[${\mathcal}{E}_{}$]{}]{}is DP-friendly, then any equivalence class of [[${\mathcal}{E}_{}$]{}]{}has a progressive representative of size at most $b({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace}) := 2^{r({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace})+1} \cdot t$, where $r({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace})$ is the function defined in Lemma \[lem:nb class\].

**An explicit protrusion replacement.** The next lemma specifies conditions under which, given an upper bound on the size of the representatives, a generic DP algorithm can provide in linear time an explicit protrusion replacer.

$[\star]$ \[lem:comput repres\] Let [${\mathcal}{G}$]{}be a class of graphs, let [[${\mathcal}{E}_{}$]{}]{}be an encoder, let $g: \mathbb{N} \rightarrow \mathbb{N}$, and let $t \in \mathbb{N}$ such that [[${\mathcal}{E}_{}$]{}]{}is $g$-confined and [[${\mathcal}{E}_{}$]{}]{}is DP-friendly. Assume we are given an upper bound $b {\geqslant}t$ on the size of a smallest progressive representative of any class of [[${\mathcal}{E}_{}$]{}]{}. Given a $t$-protrusion $Y$ inside some graph, we can compute a $t$-protrusion $Y'$ of size at most $b$ 