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. Distance hereditary graphs are exactly the graphs with rank-width at most 1. 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. It was shown by Eiben, Ganian, and Kwon (MFCS’ 16) that Distance-Hereditary Vertex Deletion can be solved in time 2O(k)nO(1), and they asked whether the problem 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 O(k) vertices when the problem is a Yes-instance, and use Mader’s S-path theorem to hit all obstructions containing exactly one vertex of the approximate solution. Then we exploit the structure of split decompositions of distance-hereditary graphs to reduce the total size. Using Mader’s S-path theorem in the context of kernelization might be of independent interest.