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

Authors Guozhen Rong, Yongjie Yang , Wenjun Li

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Guozhen Rong
  • Hunan Provincial Key Laboratory of Intelligent Processing of Big Data on Transportation, Changsha University of Science and Technology, China
Yongjie Yang
  • Chair of Economic Theory, Saarland University, Saarbrücken, Germany
Wenjun Li
  • Hunan Provincial Key Laboratory of Intelligent Processing of Big Data on Transportation, Changsha University of Science and Technology, China


The authors thank the anonymous reviewers of MFCS 2023 for their careful reading and instructive comments.

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Guozhen Rong, Yongjie Yang, and Wenjun Li. A Polynomial-Time Algorithm for MCS Partial Search Order on Chordal Graphs. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 77:1-77:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We study the partial search order problem (PSOP) proposed recently by Scheffler [WG 2022]. Given a graph G together with a partial order on the set of vertices of G, this problem determines if there is an 𝒮-ordering that is consistent with the given partial order, where 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 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Graph algorithms analysis
  • partial search order
  • maximum cardinality search
  • chordal graphs
  • clique graphs
  • dynamic programming


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