Using contracted solution graphs for solving reconfiguration problems

Abstract

We introduce in a general setting a dynamic programming method for solving reconfiguration problems. Our method is based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. Our general framework captures the approach behind known reconfiguration results of Bonsma (2012) and Hatanaka, Ito and Zhou (2014). As a third example, we apply the method to the following well-studied problem: given two k-colorings α and β of a graph G, can α be modified into β by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? This problem is known to be PSPACE-hard even for bipartite planar graphs and k = 4. By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for (k − 2)-connected chordal graphs.

The Coloring Reconfiguration Problem on Specific Graph Classes

“…More specifically, we first show that c-coloring reconfiguration remains PSPACE-complete for chordal graphs; note that c is some fixed constant. Therefore, we answer the open question posed by Bonsma and Paulusma [5]. We then demonstrate that coloring reconfiguration is solvable in polynomial time for several graph classes, even when c is a part of input; such graph classes include 2-degenerate graphs, k-trees with any integer k ≥ 1, split graphs, and trivially perfect graphs.…”

“…Coloring reconfiguration is one of the most well-studied reconfiguration problems from various viewpoints [1]- [8], [10], [15], [19], [20], including the parameterized complexity [4], [15], (in)tractability with respect to graph classes [3], [5], [19], generalized variants such as the list coloring variant [3], [10], [19], the H-coloring variant [20] and the circular coloring variant [6].…”

“…Recently, Bonsma and Paulusma [5] gave a polynomial-time algorithm to solve coloring reconfiguration for (c − 2)-connected chordal graphs; note that c is not necessarily a constant in their algorithm. They posed an open question which asks whether the problem is solvable in polynomial time for all chordal graphs.…”