$G$. This possibility prevents us from being able to prove that an encoder is DP-friendly while still being $g$-confined for some function $g$, as in order to fill in the entries of the tables at a given node, one may need to retrieve information from the tables of other nodes different from its children. To circumvent this problem, we introduce another criterion to discard the entries in the tables of an encoder: we recursively discard the entries of the tables whose associated partial solutions [*induce*]{} partial solutions at some lower node of the rooted tree decomposition that need to be discarded. That is, if an entry of the table needs to be discarded at some node of a tree decomposition, we [*propagate*]{} this information to all the other nodes.

**Organization of the paper.** Some basic preliminaries can be found in Section \[sec:prelim\], including graph minors, parameterized problems, (rooted) tree decompositions, boundaried graphs, the canonical equivalence relation $\equiv_{\Pi,t}$ for a problem $\Pi$ and an integer $t$, FII, protrusions, and protrusion decompositions. The reader not familiar with the background used in previous work on this topic may see [@BFL+09; @FLST10; @KLP+12; @KviaDP]. In Section \[sec:generic\] we introduce the basic definitions of our 