Kimelfeld and Sagiv [Kimelfeld and Sagiv, PODS 2006], [Kimelfeld and Sagiv, Inf. Syst. 2008] pointed out the problem of enumerating $K$-fragments is of great importance in a keyword search on data graphs. In a graph-theoretic term, the problem corresponds to enumerating minimal Steiner trees in (directed) graphs. In this paper, we propose a linear-delay and polynomial-space algorithm for enumerating all minimal Steiner trees, improving on a previous result in [Kimelfeld and Sagiv, Inf. Syst. 2008]. Our enumeration algorithm can be extended to other Steiner problems, such as minimal Steiner forests, minimal terminal Steiner trees, and minimal directed Steiner trees. As another variant of the minimal Steiner tree enumeration problem, we study the problem of enumerating minimal induced Steiner subgraphs. We propose a polynomial-delay and exponential-space enumeration algorithm of minimal induced Steiner subgraphs on claw-free graphs. Contrary to these tractable results, we show that the problem of enumerating minimal group Steiner trees is at least as hard as the minimal transversal enumeration problem on hypergraphs.

翻译：Kimelfeld和Sagiv, PODS 2006),[Kimelfeld和Sagiv, Inf.Syst.],[Kimelfeld和Sagiv, Inf.Syst.],[Kigelfed and Sagiv, Inf.Syst.],[Kiglefeld and Sagiv, Inf.Syst.],[Keffeld and Sagyf.Syst.],],[Kemelfflf和Sagyf.],[2008]指出,在数据图形中查找关键字搜索时,问题与数字最小的施泰纳树(方向)对应,在图形中(方向),我们建议用一个线际和多边空间算法计算所有最小的施泰纳树,改进了以前的结果,[Keqours, Instaines, 和最小方向树,最小方向, 最小方向,我们展示了作为最硬的“最高层”问题,在“最高层”图上,我们展示了“最硬的“最难的“最高树”问题是“最高树”的“最高树”检索”问题。