We study the partial search order problem (PSOP) proposed recently by Scheffler [WG 2022]. Given a graph $G$ together with a partial order over the set of vertices of $G$, this problem determines if there is an $\mathcal{S}$-ordering that is consistent with the given partial order, where $\mathcal{S}$ is a graph search paradigm like BFS, DFS, etc. This problem naturally generalizes the end-vertex problem which has received much attention over the past few years. It also generalizes the so-called ${\mathcal{F}}$-tree recognition problem which has just been studied in the literature recently. Our main contribution is a polynomial-time dynamic programming algorithm for the PSOP of the maximum cardinality search (MCS) restricted to chordal graphs. This resolves one of the most intriguing open questions left in the work of Scheffler [WG 2022]. To obtain our result, we propose the notion of layer structure and study numerous related structural properties which might be of independent interest.

翻译：我们研究了Scheffler [WG 2022]最近提出的部分搜索序问题(PSOP)。给定图$G$及其顶点集上的一个偏序，该问题判定是否存在一个与给定偏序一致的$\mathcal{S}$-排序，其中$\mathcal{S}$是像BFS、DFS等图搜索范式。这个问题自然推广了过去几年备受关注的端顶点问题，也推广了最近刚在文献中研究的所谓${\mathcal{F}}$-树识别问题。我们的主要贡献在于针对限制在弦图上的最大基数搜索(MCS)的PSOP，提出了一个多项式时间动态规划算法。这解决了Scheffler [WG 2022]工作中遗留的最具挑战性的开放问题之一。为获得该结果，我们提出了层结构的概念并研究了大量相关结构性质，这些性质可能具有独立的研究价值。