Abstract. We study the problem of transforming one list (vertex) coloring of a graph into another list coloring by changing only one vertex color assignment at a time, while at all times maintaining a list coloring, given a list of allowed colors for each vertex. This problem is known to be PSPACE-complete for bipartite planar graphs. In this paper, we first show that the problem remains PSPACE-complete even for bipartite series-parallel graphs, which form a proper subclass of bipartite planar graphs. We note that our reduction indeed shows the PSPACEcompleteness for graphs with pathwidth two, and it can be extended for threshold graphs. In contrast, we give a polynomial-time algorithm to solve the problem for graphs with pathwidth one. Thus, this paper gives precise analyses of the problem with respect to pathwidth.

Let G be a graph such that each vertex has its list of available colors, and assume that each list is a subset of the common set consisting of k colors. For two given list colorings of G, we study the problem of transforming one into the other by changing only one vertex color assignment at a time, while at all times maintaining a list coloring. This problem is known to be PSPACE-complete even for bounded bandwidth graphs and a fixed constant k. In this paper, we study the fixed-parameter tractability of the problem when parameterized by several graph parameters. We first give a fixed-parameter algorithm for the problem when parameterized by k and the modular-width of an input graph. We next give a fixed-parameter algorithm for the shortest variant when parameterized by k and the size of a minimum vertex cover of an input graph. As corollaries, we show that the problem for cographs and the shortest variant for split graphs are fixed-parameter tractable even when only k is taken as a parameter. On the other hand, we prove that the problem is W[1]-hard when parameterized only by the size of a minimum vertex cover of an input graph.Recently, the framework of reconfiguration [16] has been extensively studied in the field of theoretical computer science. This framework models several situations where we wish to find a step-by-step transformation between two feasible solutions of a combinatorial (search) problem such that all intermediate solutions are also feasible and each step respects a fixed reconfiguration rule. This reconfiguration framework has been applied to several well-studied combinatorial problems. (See a survey [15].) Our problemIn this paper, we study a reconfiguration problem for list (vertex) colorings in a graph, which was introduced by Bonsma and Cereceda [2].

Constraint satisfaction problem (CSP) is a well-studied combinatorial search problem, in which we are asked to find an assignment of values to given variables so as to satisfy all of given constraints. We study a reconfiguration variant of CSP, in which we are given an instance of CSP together with its two satisfying assignments, and asked to determine whether one assignment can be transformed into the other by changing a single variable assignment at a time, while always remaining satisfying assignment. This problem generalizes several well-studied reconfiguration problems such as Boolean satisfiability reconfiguration, vertex coloring reconfiguration, homomorphism reconfiguration. In this paper, we study the problem from the viewpoints of polynomial-time solvability and parameterized complexity, and give several interesting boundaries of tractable and intractable cases.

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We study the problem of transforming one (vertex) ccoloring of a graph into another one by changing only one vertex color assignment at a time, while at all times maintaining a c-coloring, where c denotes the number of colors. This decision problem is known to be PSPACE-complete even for bipartite graphs and any fixed constant c ≥ 4. In this paper, we study the problem from the viewpoint of graph classes. We first show that the problem remains PSPACE-complete for chordal graphs even if c is a fixed constant. We then demonstrate that, even when c is a part of input, the problem is solvable in polynomial time for several graph classes, such as k-trees with any integer k ≥ 1, split graphs, and trivially perfect graphs.