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

TL;DR

This paper presents a polynomial-time dynamic programming algorithm for solving the partial search order problem for maximum cardinality search on chordal graphs, addressing an open question in graph search algorithms.

Contribution

The paper introduces a novel polynomial-time algorithm for the PSOP of MCS on chordal graphs, resolving a key open problem in the field.

Findings

Developed a polynomial-time dynamic programming algorithm for PSOP of MCS on chordal graphs.

Introduced the concept of layer structure and studied related structural properties.

Resolved an open question from Scheffler's 2022 work.

Abstract

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.