A Polynomial-Time Algorithm for MCS Partial Search Order on Chordal Graphs

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 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 on chordal graphs with respect to the maximum cardinality search (MCS). This resolves one of the most intriguing open questions left in the work of Sheffler [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.