Let $G'\in {\ensuremath{{\mathcal}{B}_t}\xspace}$ with $\partial(G') = A$ be the graph obtained from $G$ by replacing the subgraph $G_B$ with some $G_B' \in {\ensuremath{{\mathcal}{B}_t}\xspace}$ such that $G_B {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace}^*}\xspace} G_B'$. Then $G {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace}^*}\xspace} G'$ and ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}(G,G') = {\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}(G_B, G_B')$.

The following useful fact states that for proving that [[${\mathcal}{E}_{}$]{}]{}is DP-friendly, it suffices to prove that $G {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}^*}\xspace} G'$ instead of $G {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace}^*}\xspace} G'$.

$[\star]$ \[fait:equiv\] Let $G\in {\ensuremath{{\mathcal}{B}_t}\xspace}$ with a separator $B$, let $G_B {\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace} G_B'$, and let $G' \in {\ensuremath{{\mathcal}{B}_t}\xspace}$ as in Definition \[defi:DP-friend\]. If $G {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}^*}\xspace} G'$, then $G {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace}^*}\xspace} G'$.

In order to perform a protrusion replacement that does not modify the behavior of the graph with respect to a problem $\Pi$, we need the relation [[${\mathcal}{E}_{}$]{}]{}to be a refinement of the canonical equivalence relation $\equiv_{\Pi,t}$.

$[\star]$\[lem:refine eq\] Let $\Pi$ be a packing-certifiable parameterized problem defined on a graph class ${\cal G}$, let [[${\mathcal}{E}_{}$]{}]{}be an encoder, let $g: \mathbb{N} \rightarrow \mathbb{N}$, and let $G_1, G_2 \in {\ensuremath{{\mathcal}{B}_t}\xspace}$. If [[${\mathcal}{E}_{}$]{}]{}is a $g$-confined $\Pi$-encoder and [[${\mathcal}{E}_{}$]{}]{}is DP-friendly, then the fact that $G_1 {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace}^*}\xspace} G_2$ implies the following:

-   $G_1 \equiv_\Pi G_2$; and

-   $\Delta_{\Pi,t}(G_1,G_2) = {\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}(G_1,G_2)$.

In particular, this holds when $G_1, G_2 \in {\ensuremath{{\mathcal}{F}_t}\xspace}$ and $G_1 {\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace} G_2$.

\[defi:progres\] Given 