appeared in the last years, starting with the result of Bodlaender *et al*. [@BFL+09] on graphs of bounded genus. It was followed-up by similar results on larger sparse graph classes, such as graphs excluding a minor [@FLST10] or a topological minor [@KLP+12].

The above results guarantee the [*existence*]{} of linear kernels on sparse graph classes for problems satisfying some generic conditions, but it is hard to derive from them [*constructive*]{} kernels with [*explicit*]{} constants. We recently made in [@KviaDP] a significant step toward a fully constructive meta-kernelization theory on sparse graphs with explicit constants. In a nutshell, the main idea is to substitute the algorithmic power of CMSO logic that was used in [@BFL+09; @FLST10; @KLP+12] with that of dynamic programming (DP for short) on graphs of bounded decomposability (i.e., bounded treewidth). We refer the reader to the introduction of [@KviaDP] for more details. Our approach provides a DP framework able to construct linear kernels for families of problems on sparse graphs whose solutions can be certified by a subset of [*vertices*]{} of the input graph, such as $r$-<span style="font-variant:small-caps;">Dominating Set</span> or <span style="font-variant:small-caps;">Planar-$\mathcal{F}$-Deletion</span>.

**Our contribution.** In this article we make one more step in the direction of a fully 