with $O((r+1)^r k^r)$ vertices, where $r$ is the maximum number of vertices of a graph in $\mathcal{F}$. In this article we improve this result on graphs excluding a fixed graph as a minor, by providing a linear kernel for <span style="font-variant:small-caps;">$\mathcal{F}$-Subgraph-Packing with $\ell$-Membership</span> when $\mathcal{F}$ is any family of (not necessarily planar) connected graphs.

**Our techniques: vertex-certifiable vs. packing-certifiable problems**. It appears that packing-certifiable problems are intrinsically more involved than vertex-certifiable ones. This fact is well-known when speaking about [[FPT]{}]{}-algorithms on graphs of bounded treewidth [@CyganNPPRW11; @LokshtanovMS11], but we need to be more precise with what we mean by being “more involved” in our setting of obtaining kernels via DP on a tree decomposition of the input graph. Loosely speaking, the framework that we presented in [@KviaDP] and that we need to redefine and extend here, can be summarized as follows. First of all, we propose a general definition of a problem *encoding* for the tables of DP when solving parameterized problems on graphs of bounded treewidth. Under this setting, we provide three general conditions guaranteeing that such an encoding can yield a so-called *protrusion replacer*, which in short is a procedure that replaces large “protrusions” (i.e., subgraphs with small 