The graph searches Breadth First Search (BFS) and Depth First Search (DFS) and the spanning trees constructed by them are some of the most basic concepts in algorithmic graph theory. BFS trees are first-in trees, i.e., every vertex is connected to its first visited neighbor. DFS trees are last-in trees, i.e., every vertex is connected to the last visited neighbor before it. The problem whether a given spanning tree can be the first-in tree or last-in tree of a graph search ordering was introduced in the 1980s and has been studied for several graph searches and graph classes. Here, we consider the problem of deciding whether a given spanning tree of a bipartite graph can be a first-in tree or a last-in tree of the Lexicographic Breadth First Search (LBFS), a special variant of BFS that is commonly used in graph algorithms. We show that the recognition of both first-in trees and last-in trees of LBFS is -hard even if the start vertex of the search ordering is fixed and the height of the tree is four. We prove that the bound on the height is tight (unless ) by showing that for all spanning trees of bipartite graphs with height smaller than four we can solve both search tree recognition problems of LBFS in polynomial time. Finally, we give a linear-time algorithm that solves both problems for chordal bipartite graphs and fixed start vertices.

中文翻译：

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识别二分图的 LBFS 树

图搜索广度优先搜索（BFS）和深度优先搜索（DFS）以及它们构造的生成树是算法图论中的一些最基本的概念。BFS 树是先进树，即每个顶点都连接到其第一个访问的邻居。DFS 树是后进树，即每个顶点都连接到它之前最后访问的邻居。给定的生成树是否可以是图搜索排序的先入树或后入树的问题是在 20 世纪 80 年代引入的，并且已经针对多个图搜索和图类进行了研究。在这里，我们考虑确定二分图的给定生成树是否可以是词典广度优先搜索（LBFS）的先进树或后入树的问题，LBFS是BFS的一种特殊变体，常用于图算法。我们表明，即使搜索顺序的起始顶点是固定的并且树的高度为 4，LBFS 的先入树和后入树的识别也是困难的。我们通过证明对于高度小于四的二分图的所有生成树，我们可以在多项式时间内解决 LBFS 的搜索树识别问题，证明高度上的界限是紧的（除非 ）。最后，我们给出了一种线性时间算法，可以解决弦二部图和固定起始顶点的问题。