Bridge-Depth Characterizes which Minor-Closed Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel

We study the kernelization complexity of structural parameterizations Vertex Cover problem. Here, goal is to find a polynomial-time preprocessing algorithm that can reduce any instance $(G,k)$ problem an equivalent one, whose size polynomial in pre-determined parameter $G$. A long line previous research deals with based on number vertex deletions needed $G$ member simple graph class $\mathcal{F}$, such as forests, graphs bounded tree-depth, and maximum degree two. set out most general classes $\mathcal{F}$ for which parameterized by vertex-deletion distance input admits kernelization. give complete characterization minor-closed families exists. introduce new called bridge-depth, prove exists if only has bridge-depth. The proof interesting connection between bridge-depth minimal blocking sets graphs, are removal decreases independence number.