\[defi:equiv\]). Following the notation of Bodlaender *et al*. [@BFL+09], the function ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}$ is called the *transposition function* for the equivalence relation [[${\mathcal}{E}_{}$]{}]{}. Note that we can use the restriction of ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}$ to couples of graphs in [${\mathcal}{F}_t$]{}to define the equivalence relation [[${\mathcal}{E}_{}$]{}]{}.

In the following we only consider classes of graphs whose membership can be expressed in Monadic Second Order (MSO) logic. Therefore, we know that the number of equivalence classes of [${\mathcal}{G}$]{}is finite [@Buc60], say at most $r_{{\ensuremath{{\mathcal}{G}}\xspace},t}$, and we can state the following lemma.

$[\star]$ \[lem:nb class\] 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 integer $t \in \mathbb{N}$, if [[${\mathcal}{E}_{}$]{}]{}is $g$-confined then the equivalence relation [[${\mathcal}{E}_{}$]{}]{}has at most $ r({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace}):=(g(t)+2)^{s_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}(t)} \cdot 2^t \cdot r_{{\ensuremath{{\mathcal}{G}}\xspace},t} $ equivalence classes. In particular, the equivalence relation [[${\mathcal}{E}_{}$]{}]{}has at most $ r({{\ensuremath{{\mathcal}{E}_{}}\xspace}},g,t,{\ensuremath{{\mathcal}{G}}\xspace})$ equivalence classes as well.

\[defi:DP-friend\] An equivalence relation [[${\mathcal}{E}_{}$]{}]{}is *DP-friendly* if, for any graph $G \in {\ensuremath{{\mathcal}{B}_t}\xspace}$ with $\partial(G) = A$ and any two boundaried graphs $H$ and $G_B$ with $G= H \oplus G_B$ such that $G_B$ has boundary $B \subseteq V(G)$ with $|B| {\leqslant}t$ and $A\cap V(G_B) \subseteq B$, the following holds. 