A Polynomial Kernel for Distance-Hereditary Vertex Deletion

A graph is distance-hereditary if for any pair of vertices, their distance in every connected induced subgraph containing both vertices is the same as their distance in the original graph. The Distance-Hereditary Vertex Deletion problem asks, given a graph $G$ on $n$ vertices and an integer $k$, whether there is a set $S$ of at most $k$ vertices in $G$ such that $G-S$ is distance-hereditary. This problem is important due to its connection to the graph parameter rank-width that distance-hereditary graphs are exactly graphs of rank-width at most $1$. Eiben, Ganian, and Kwon (MFCS' 16) proved that Distance-Hereditary Vertex Deletion can be solved in time $2^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$, and asked whether it admits a polynomial kernelization. We show that this problem admits a polynomial kernel, answering this question positively. For this, we use a similar idea for obtaining an approximate solution for Chordal Vertex Deletion due to Jansen and Pilipczuk (SODA' 17) to obtain an approximate solution with $\mathcal{O}(k^3\log n)$ vertices when the problem is a YES-instance, and we exploit the structure of split decompositions of distance-hereditary graphs to reduce the total size.