Given a simple connected undirected graph G = (V, E), a set X \subseteq V(G), and integers k and p, STEINER SUBGRAPH EXTENSION problem asks if there exists a set S \supseteq X with at most k vertices such that G[S] is p-edge-connected. This is a natural generalization of a well-studied problem STEINER TREE (set p=1 and X as the set of all terminals). In this paper, we initiate the study of STEINER SUBGRAPH EXTENSION from the perspective of parameterized complexity and give a fixed-parameter algorithm parameterized by k and p on graphs of bounded degeneracy. In case we remove the assumption of the input graph being bounded degenerate, then the STEINER SUBGRAPH EXTENSION problem becomes W[1]-hard. Besides being an independent advance on the parameterized complexity of network design problems, our result has natural applications. In particular, we use our result to obtain singly exponential-time FPT algorithms for several vertex deletion problem studied in the literature, where the goal is to delete a smallest set of vertices such that (i) the resulting graph belongs to a specific hereditary graph class, and (ii) the deleted set of vertices induces a p-edge-connected subgraph of the input graph.

翻译：给定一个简单连通无向图 G = (V, E)、一个顶点子集 X ⊆ V(G) 以及整数 k 和 p，斯坦纳子图扩展问题询问是否存在一个顶点集 S ⊇ X，其大小至多为 k，使得诱导子图 G[S] 是 p-边连通的。这是一个已被深入研究的斯坦纳树问题的自然推广（令 p=1 并将 X 设为所有终端的集合）。本文从参数化复杂度的角度首次对斯坦纳子图扩展问题进行研究，并在有界退化图上给出了一个以 k 和 p 为参数的固定参数可解算法。若移除输入图具有有界退化性的假设，则斯坦纳子图扩展问题变为 W[1]-困难的。除了在网络设计问题的参数化复杂度方面取得独立进展外，我们的结果具有自然的应用价值。具体而言，我们利用该结果为文献中研究的若干顶点删除问题获得了单指数时间的 FPT 算法，这些问题的目标是删除一个最小的顶点集合，使得 (i) 删除后剩余的图属于某个特定的遗传图类，且 (ii) 被删除的顶点集在输入图中诱导出一个 p-边连通的子图。