Reconfiguration of Colorable Sets in Classes of Perfect Graphs

Authors Takehiro Ito , Yota Otachi

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Takehiro Ito
  • Graduate School of Information Sciences, Tohoku University. , Aoba-yama 6-6-05, Sendai, 980-8579, Japan
Yota Otachi
  • Faculty of Advanced Science and Technology, Kumamoto University. , 2-39-1 Kurokami, Chuo-ku, Kumamoto, 860-8555 Japan

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Takehiro Ito and Yota Otachi. Reconfiguration of Colorable Sets in Classes of Perfect Graphs. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (reconfiguration) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. This gives a linear-time algorithm for finding an actual shortest reconfiguration sequence for interval graphs. Since interval graphs are exactly the graphs that are simultaneously chordal and co-comparability, we then complement the positive result by showing that even deciding reachability is PSPACE-complete for chordal graphs and for co-comparability graphs. The hardness for chordal graphs holds even for split graphs. We also consider the case where c is a fixed constant and show that in such a case the reachability problem is polynomial-time solvable for split graphs but still PSPACE-complete for co-comparability graphs. The complexity of this case for chordal graphs remains unsettled. As by-products, our positive results give the first polynomial-time solvable cases (split graphs and interval graphs) for Feedback Vertex Set Reconfiguration.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • reconfiguration
  • colorable set
  • perfect graph


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