Graph searches and their respective search trees are widely used in algorithmic graph theory. The problem whether a given spanning tree can be a graph search tree has been considered for different searches, graph classes and search tree paradigms. Similarly, the question whether a particular vertex can be visited last by some search has been studied extensively in recent years. We combine these two problems by considering the question whether a vertex can be a leaf of a graph search tree. We show that for particular search trees, including DFS trees, this problem is easy if we allow the leaf to be the first vertex of the search ordering. We contrast this result by showing that the problem becomes hard for many searches, including DFS and BFS, if we forbid the leaf to be the first vertex. Additionally, we present several structural and algorithmic results for search tree leaves of chordal graphs.

翻译：图搜索及其对应的搜索树在算法图论中被广泛使用。对于不同搜索方式、图类和搜索树范式，已开展关于给定生成树能否作为图搜索树的研究。类似地，特定顶点能否在某种搜索中最后被访问的问题近年来也受到广泛关注。我们通过考虑顶点能否成为图搜索树叶子节点的问题，将上述两个问题结合起来。研究表明，对于包括DFS树在内的特定搜索树，若允许该叶子节点作为搜索顺序的首个顶点，则问题易于求解。与此形成对比的是，若禁止该叶子节点作为首个顶点，则对于包括DFS和BFS在内的多种搜索方式，问题将变得困难。此外，我们还针对弦图的搜索树叶子节点给出了若干结构性和算法性结果。