SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 1418-1446, January 2021.
Graph searches and the corresponding search trees can exhibit important structural properties and are used in various graph algorithms. The problem of deciding whether a given spanning tree of a graph is a search tree of a particular search on this graph was introduced by Hagerup in 1985, where the author showed that this problem is efficiently solvable for depth first search (DFS) trees and breadth first search (BFS) trees. If one defines such a search tree in the same way as done for BFS, i.e., by connecting every vertex to its first neighbor, then we call this an ${\cal F}$-tree. If, on the other hand, we connect it with its most recently visited neighbor (as in DFS) we call this an ${\cal L}$-tree. In this paper, we consider related search paradigms. We prove that the search tree problem can be solved in polynomial time for ${\cal L}$-trees of lexicographic depth first search, whereas the ${\cal F}$-tree recognition problem is $\mathcal{NP}$-complete for lexicographic breadth first search, lexicographic depth first search, maximum cardinality search, and maximal neighborhood search. Furthermore, we present polynomial results for both types of trees on chordal graphs.

中文翻译：

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图搜索树的识别问题

SIAM 离散数学杂志，第 35 卷，第 2 期，第 1418-1446 页，2021 年 1 月。
图搜索和相应的搜索树可以表现出重要的结构特性，并用于各种图算法。Hagerup 在 1985 年引入了决定图的给定生成树是否是该图上特定搜索的搜索树的问题，作者表明该问题对于深度优先搜索 (DFS) 树和广度是可有效解决的第一搜索（BFS）树。如果以与 BFS 相同的方式定义这样的搜索树，即通过将每个顶点连接到它的第一个邻居，那么我们称其为 ${\cal F}$-tree。另一方面，如果我们将它与最近访问过的邻居（如在 DFS 中）连接起来，我们称其为 ${\cal L}$-tree。在本文中，我们考虑相关的搜索范式。我们证明搜索树问题可以在多项式时间内解决字典深度优先搜索的${\cal L}$-trees，而${\cal F}$-tree 识别问题是$\mathcal{NP}$ - 完成词典广度优先搜索、词典深度优先搜索、最大基数搜索和最大邻域搜索。此外，我们在弦图上展示了两种类型的树的多项式结果。