We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight spanning tree among those containing a perfect matching. On the positive side, we design a simple greedy algorithm for the case when the graph is complete (or complete bipartite) and the edge weights take at most two values. On the negative side, the problem is NP-hard even when the graph is complete (or complete bipartite) and the edge weights take at most three values, or when the graph is cubic, planar, and bipartite and the edge weights take at most two values. We also consider an interesting variant. We call a tree strongly balanced if on one side of the bipartition of the vertex set with respect to the tree, all but one of the vertices have degree $2$ and the remaining one is a leaf. This property is a sufficient condition for a tree to have a perfect matching, which enjoys an additional property. When the underlying graph is bipartite, strongly balanced spanning trees can be written as matroid intersection, and this fact was recently utilized to design an approximation algorithm for some kind of connectivity augmentation problem. The natural question is its tractability in nonbipartite graphs. As a negative answer, it turns out NP-hard to test whether a given graph has a strongly balanced spanning tree or not even when the graph is subcubic and planar.

翻译：我们研究了图论中两个基本结构——生成树与完美匹配——简单融合的可处理性。具体而言，我们考虑以下问题：给定一个边赋权图，在所有包含完美匹配的生成树中寻找权重最小者。在积极方面，我们针对图是完全图（或完全二分图）且边权至多取两个值的情形，设计了一种简单的贪心算法。在消极方面，即使图是完全图（或完全二分图）且边权至多取三个值，或者图是三次、平面且二分的且边权至多取两个值，该问题也是NP难的。我们还考虑了一个有趣的变体。若在树所对应的顶点集二分划分的某一侧，除一个顶点度为$2$且余下顶点均为叶子节点，则称该树为强平衡树。此性质是树具有完美匹配的充分条件，且该完美匹配还满足一个额外性质。当底层图为二分图时，强平衡生成树可表述为拟阵交问题，这一事实最近被用于设计某类连通性增强问题的近似算法。一个自然的问题是其在非二分图中的可处理性。作为否定回答，即使图是次三次且平面的，判定给定图是否存在强平衡生成树也是NP难的。