encoder [[${\mathcal}{E}_{}$]{}]{}and a function $g$, in the next sections we will denote the [*relevant*]{} functions discussed before by ${\ensuremath{\bar f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}_{g}}\xspace}$ to distinguish them from other functions that we will need.

**Equivalence relations and representatives.** We now define some equivalence relations on $t$-boundaried graphs.

\[defi:equiv\] Let [[${\mathcal}{E}_{}$]{}]{}be an encoder, let $G_1,G_2 \in {\ensuremath{{\mathcal}{B}_t}\xspace}$, and let ${\ensuremath{{\mathcal}{G}}\xspace}$ be a class of graphs.

1.  $G_1 {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}^*}\xspace} G_2$ if $\Lambda(G_1)=\Lambda(G_2)=: I$ and there exists an integer ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}(G_1,G_2)$ (depending on $G_1, G_2$) such that for any encoding $R   \in {\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(I)$ we have ${\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G_1,R) = {\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G_2,R) - {\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace} (G_1,G_2)$.

2.  $G_1 {\ensuremath{\sim_{{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace} G_2$ if either $G_1 \notin {\ensuremath{{\mathcal}{G}}\xspace}$ and $G_2 \notin {\ensuremath{{\mathcal}{G}}\xspace}$, or $G_1,G_2 \in {\ensuremath{{\mathcal}{G}}\xspace}$ and, for any $H \in {\ensuremath{{\mathcal}{B}_t}\xspace}$, $H \oplus G_1 \in {\ensuremath{{\mathcal}{G}}\xspace}$ if and only if $H \oplus G_2 \in {\ensuremath{{\mathcal}{G}}\xspace}$.

3.  $G_1 {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace}^*}\xspace} G_2$ if $G_1 {\ensuremath{{\ensuremath{\sim_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace}^*}\xspace} G_2$ and $G_1 {\ensuremath{\sim_{{\ensuremath{{\mathcal}{G}}\xspace},t}}\xspace} G_2$.

4.  If we restrict the graphs $G_1 , G_2$ to be in [${\mathcal}{F}_t$]{}, then the corresponding equivalence relations, which are a restriction of [[${\mathcal}{E}_{}$]{}]{}and [[${\mathcal}{E}_{}$]{}]{}, are denoted by [[${\mathcal}{E}_{}$]{}]{}and [[${\mathcal}{E}_{}$]{}]{}, respectively.

If for all encodings $R$, ${\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G_1,R) = {\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G_2,R) = -\infty$, then we set ${\ensuremath{\Delta_{{{\ensuremath{{\mathcal}{E}_{}}\xspace}},t}}\xspace} (G_1,G_2):=0$ (note that any fixed integer would satisfy the first condition in Definition 