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 the same as original graph. The Distance-Hereditary Vertex Deletion problem asks, given a G on n and an integer k, whether there set S at most k such that $$G-S$$ distance-hereditary. This important due to its connection parameter rank-width because graphs are exactly 1. Eiben, Ganian, Kwon (JCSS’ 18) proved can be solved time $$2^{{\mathcal {O}}(k)}n^{{\mathcal {O}}(1)}$$ , asked it admits polynomial kernelization. We show this kernel, answering question positively. For this, we use similar idea obtaining approximate solution Chordal Jansen Pilipczuk (SIDMA’ obtain with $${\mathcal {O}}(k^3\log n+ k^2\log ^2 n)$$ when Yes-instance, exploit structure split decompositions reduce total size.