Let G = (V, E) be a connected graph and X be a vertex subset of G. Let f be a mapping from X to the set of natural numbers such that fix) ≥ 2 for all x ∈ X. A degree restricted spanning tree is a spanning tree T of G such that f(X) ≤ deg_T(x) for all x ∈ X, where deg_T(x) denotes the degree of a vertex x in T. In this paper, we show that the decision problem "whether there exists a degree restricted spanning tree in G" is NP-complete. We also give a restricted proof of a conjecture, provided by Kaneko and Yoshimoto, on the existence of such a spanning tree in general graphs. Finally, we present a polynomial-time algorithm to find a degree restricted spanning tree of a graph satisfying the conditions presented in the restricted proof of the conjecture.
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机译：设g =（v，e）是连接的图形，x是g的顶点子集。让f是从x到的映射到一个自然数，这样的固定）≥2对于所有x x x x x x。程度受限制 树是G的生成树T，使得所有x∈X的f（x）≤deg_t（x），其中deg_t（x）表示T的顶点X的程度。在本文中，我们展示了决策问题 “在G中是否存在受限制的生成树”是NP-Tress。 我们还提供了由Kaneko和Yoshimoto提供的猜想的限制证明，在一般图中存在这样的生成树。 最后，我们提出了一种多项式时间算法，用于找到满足猜测限制证明中所呈现的条件的图的程度受限的跨度树。
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