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 [19]; distance-hereditary graphs are exactly the 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.