AI 中文总结
本文解决半完全有向图中端点分属不同强分量的最长(x,y)-路径问题，确定局部半完全有向图连通非强情形的可能端点，给出多项式时间精确算法并分析未解决有向图的性质。
AI 中文摘要
我们研究Bang-Jensen和Gutin提出的两个带指定端点的开放路径问题。第一个问题是求半完全有向图中的最长(x,y)-路径；第二个问题是判断局部半完全有向图是否存在哈密顿(x,y)-路径。对于半完全有向图，当端点位于不同强分量时，我们解决了第一个问题。我们还证明，若非哈密顿的最长(x,y)-路径遗漏了一组顶点，则这些顶点与x、y共同构成的集合存在哈密顿(y,x)-路径，这等价于一个循环问题。随后，我们给出了一个多项式时间的精确算法，当遗漏顶点的数量固定时可运行。对于局部半完全有向图，我们确定了连通非强情形下的可能端点。已知结果仅剩下强、非半完全、非4-强有向图未解决，每个阶至少为5的此类图都有大小至多为3的强顶点割。两个例子表明，生成有向路径加上顶点不相交的有向循环是不够的，且一个强分量的顶点在哈密顿路径上不必连续出现。
英文摘要
We study two open path problems with prescribed endpoints posed by Bang-Jensen and Gutin. The first asks for a longest $(x,y)$-path in a semicomplete digraph. The second asks whether a locally semicomplete digraph has a Hamiltonian $(x,y)$-path. For semicomplete digraphs, we solve the first problem when the endpoints lie in different strong components. We also prove that if a non-Hamiltonian longest $(x,y)$-path omits a set of vertices, then these vertices together with $x$ and $y$ have a Hamiltonian $(y,x)$-path. This gives an equivalent cycle problem. We then give an exact algorithm that runs in polynomial time when the number of omitted vertices is fixed. For locally semicomplete digraphs, we determine the possible endpoints in the connected nonstrong case. Known results then leave only strong, nonsemicomplete, non-$4$-strong digraphs unresolved. Every such digraph of order at least five has a strong vertex cut of size at most three. Two examples show that a spanning directed path together with a vertex-disjoint directed cycle is not sufficient, and that vertices of one strong component need not occur consecutively on a Hamiltonian path.
CommentsTwo of the results of the paper, Theorems 1.6 and 1.7, follow rather easily from other conclusions, so these two results are trivial
