In recent years, questions about the construction of special orderings of a given graph search were studied by several authors. On the one hand, the so called end-vertex problem introduced by Corneil et al. in 2010 asks for search orderings ending in a special vertex. On the other hand, the problem of finding orderings that induce a given search tree was introduced already in the 1980s by Hagerup and received new attention most recently by Beisegel et al. Here, we introduce a generalization of some of these problems by studying the question whether there is a search ordering that is a linear extension of a given partial order on a graph's vertex set. We show that this problem can be solved in polynomial time on chordal bipartite graphs for LBFS, which also implies the first polynomial-time algorithms for the end-vertex problem and two search tree problems for this combination of graph class and search. Furthermore, we present polynomial-time algorithms for LBFS and MCS on split graphs which generalize known results for the end-vertex and search tree problems.
翻译:近些年来,一些作者研究了关于某一图类搜索特殊命令的构建问题。一方面,Corneil等人在2010年提出的所谓终端顶部问题要求搜索以特殊的顶部结束。另一方面,哈杰鲁普在1980年代已经提出并最近得到Beisegel等人的新的关注,寻找导致某一搜索树的订单的问题已在1980年代提出,我们在这里对其中一些问题进行了概括。我们通过研究以下问题,对其中的一些问题进行了概括化介绍:是否有一个搜索命令是某一图类部分命令的线性扩展。我们表明,这个问题可以在LBFS的CBdal双部分图解的多元时间中解决,这也意味着对最终的顶部问题首次采用多元时间算法,以及这种图形类和搜索组合的两种搜索树问题。此外,我们提出了LBFS和MCS的多元图解式算法,这些图解概括了末脊部和搜索树问题的已知结果。