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(O(k)) n(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 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.