TY - JOUR

T1 - Reconfiguration of colorable sets in classes of perfect graphs

AU - Ito, Takehiro

AU - Otachi, Yota

N1 - Publisher Copyright:
© 2018 Elsevier B.V.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/6/7

Y1 - 2019/6/7

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 (called a reconfiguration sequence) 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 (called a reconfiguration sequence) 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 - Combinatorial reconfiguration

KW - Graph algorithm

KW - Perfect graph

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U2 - 10.1016/j.tcs.2018.11.024

DO - 10.1016/j.tcs.2018.11.024

M3 - Article

AN - SCOPUS:85057536164

VL - 772

SP - 111

EP - 122

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -