treewidth and small boundary) with “equivalent” subgraphs of constant size. Let us be more concrete on these three conditions that such an encoding ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ needs to satisfy in order to obtain an explicit linear kernel for a parameterized problem $\Pi$. The first natural condition is that on a graph $G$ without boundary, the optimal size of the objects satisfying the constraints imposed by ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ coincides with the optimal size of solutions of $\Pi$ in $G$; in that case we say that ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ is a *$\Pi$-encoder*. On the other hand, we need that when performing DP using the encoding ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$, we can use tables such that the maximum difference among all the values that need to be stored is bounded by a function $g$ of the treewidth; in that case we say that ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ is *$g$-confined*. Finally, the third condition requires that ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ is “suitable” for performing DP, in the sense that the tables at a given node of a tree decomposition can be computed using only the information stored in the tables of its children (as it is the case of practically all natural DP algorithms); in that case we say that ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ is *DP-friendly*. These two latter properties exhibit 