TY - GEN

T1 - Reconfiguration of colorable sets in classes of perfect graphs

AU - Ito, Takehiro

AU - Otachi, Yota

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

AB - 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.

KW - Colorable set

KW - Perfect graph

KW - Phrases reconfiguration

UR - http://www.scopus.com/inward/record.url?scp=85049053274&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049053274&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SWAT.2018.27

DO - 10.4230/LIPIcs.SWAT.2018.27

M3 - Conference contribution

AN - SCOPUS:85049053274

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 271

EP - 2713

BT - 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018

A2 - Eppstein, David

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018

Y2 - 18 June 2018 through 20 June 2018

ER -