Spanning trees are fundamental for efficient communication in networks. For fault-tolerant communication, it is desirable to have multiple spanning trees to ensure resilience against failures of nodes and edges. To this end, various notions of disjoint or independent spanning trees have been studied, including edge-disjoint, node/edge-independent, and completely independent spanning trees. Alongside these, several Steiner variants have also been investigated, where the trees are required to span a designated subset of vertices called terminals. For instance, the study of edge-disjoint spanning trees has been extended to edge-disjoint Steiner trees; a stronger variant is the problem of internally disjoint Steiner trees, where any two Steiner trees intersect exactly in the terminals. In this paper, we investigate the Steiner analogue of completely independent spanning trees, which we call \emph{completely independent Steiner trees}. A set of Steiner trees is completely independent if, for every pair of terminals $u,v$, the $(u,v)$-paths in all the Steiner trees are internally vertex-disjoint and edge-disjoint. This notion generalizes both completely independent spanning trees and internally disjoint Steiner trees. We provide a systematic study of completely independent Steiner trees from structural, algorithmic, and complexity-theoretic perspectives. In particular, we present several characterisations, connectivity bounds, algorithms, hardness results, and applications to special graph classes such as planar graphs and graphs of bounded treewidth. Along the way, we also introduce a directed variant of completely independent spanning trees via an equivalence with completely independent Steiner trees.

翻译：生成树是网络中高效通信的基础。为保障容错通信，需要多棵生成树以确保在节点和边故障时仍具韧性。为此，研究者提出了多种不相交或独立生成树的概念，包括边不相交、节点/边独立以及完全独立生成树。此外，还探索了若干斯坦纳变体，要求这些树生成称为终端的指定顶点子集。例如，边不相交生成树的研究已推广至边不相交斯坦纳树；其更强变体是内部不相交斯坦纳树问题，即任意两棵斯坦纳树仅在终端处相交。本文研究了完全独立生成树的斯坦纳类比，称为**完全独立斯坦纳树**。一组斯坦纳树若满足：对任意一对终端$u,v$，所有斯坦纳树中的$(u,v)$路径在内部顶点和边上均不相交，则称其为完全独立。该概念同时推广了完全独立生成树和内部不相交斯坦纳树。我们从结构、算法和复杂性理论角度对完全独立斯坦纳树进行了系统研究，具体包括：多种刻画、连通性界、算法、难度结果及其在平面图和有界树宽图等特殊图类上的应用。在此过程中，我们还通过完全独立斯坦纳树与完全独立生成树的等价性，引入了后者的有向变体。