A Polynomial Kernel for Distance-Hereditary Vertex Deletion

Abstract

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 because distance-hereditary graphs are exactly the graphs of rank-width at most 1. Eiben, Ganian, and Kwon (JCSS’ 18) 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 (SIDMA’ 18) to obtain an approximate solution with ${\mathcal {O}}(k^3\log n+ k^2\log ^2 n)$ 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.