  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.
