Abstract
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.
Highlights
The reconfiguration framework has been applied to several search problems
We show that Colorable Set Reconfiguration under TAR/TJ is linear-time solvable on interval graphs
What is the complexity of CSRTAR with c = 2 for chordal graphs? This problem is equivalent to the reconfiguration of feedback vertex sets under TAR on chordal graphs
Summary
The reconfiguration framework has been applied to several search problems. In a reconfiguration problem, we are given two feasible solutions of a search problem and are asked to determine whether we can modify one to the other by repeatedly applying prescribed reconfiguration rules while keeping the feasibility (see [14, 22, 19]). This property often makes problems related to coloring tractable To understand this very general problem, we start the study of Colorable Set Reconfiguration on classes of perfect graphs. We first study the problem on interval graphs and show that a shortest reconfiguration sequence under TAR can be found in linear time (Theorem 3.11) This implies the same result under TJ. As a byproduct of Theorems 3.11 and 4.4, the Feedback Vertex Set Reconfiguration problem [18] turns out to be polynomial-time solvable for split graphs and interval graphs under TAR and TJ. To see the polynomial-time solvability, observe that the complements V (G) \ S of 2-colorable sets S in a chordal graph G are exactly the feedback vertex sets in the graph and reconfigurations of the complements are equivalent to reconfigurations of the original vertex sets under TAR and TJ
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