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.

如果对于任何一对顶点，图在包含两个顶点的每个相连的诱导子图中的距离与原始图中的距离相同，则该图是距离遗传的。的距离-遗传顶点删除问题询问，给定图ģ上Ñ顶点和的整数ķ，是否有一组小号至多ķ在顶点ģ使得\（GS \）是距离遗传性。这个问题很重要，因为它与图形参数的rank-width有关，因为距离遗传图恰好是最多1个rank-width的图。Eiben，Ganian和Kwon（JCSS'18）证明了距离遗传顶点删除可以在时间\（2 ^ {{\ mathcal {O}}（k）} n ^ {{\ mathcal {O}}（1）} \）中求解，并询问是否允许多项式内核化。我们表明，该问题允许多项式核，对这一问题的回答是肯定的。为此，我们使用类似的思想来获得由于Jansen和Pilipczuk（SIDMA'18）导致的弦顶点删除的近似解，从而获得具有\（{\ mathcal {O}}（k ^ 3 \ log n + k当问题为是实例时，^ 2 \ log ^ 2 n）\）顶点，并且我们利用距离遗传图的拆分分解结构来减小总大小。