Preprocessing Vertex-Deletion Problems: Characterizing Graph Properties by Low-Rank Adjacencies

We consider the $\Pi$-free Deletion problem parameterized by the size of a vertex cover, for a range of graph properties $\Pi$. Given an input graph $G$, this problem asks whether there is a subset of at most $k$ vertices whose removal ensures the resulting graph does not contain a graph from $\Pi$ as induced subgraph. Many vertex-deletion problems such as Perfect Deletion, Wheel-free Deletion, and Interval Deletion fit into this framework. We introduce the concept of characterizing a graph property $\Pi$ by low-rank adjacencies, and use it as the cornerstone of a general kernelization theorem for $\Pi$-Free Deletion parameterized by the size of a vertex cover. The resulting framework captures problems such as AT-Free Deletion, Wheel-free Deletion, and Interval Deletion. Moreover, our new framework shows that the vertex-deletion problem to perfect graphs has a polynomial kernel when parameterized by vertex cover, thereby resolving an open question by Fomin et al. [JCSS 2014]. Our main technical contribution shows how linear-algebraic dependence of suitably defined vectors over $\mathbb{F}_2$ implies graph-theoretic statements about the presence of forbidden induced subgraphs.