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Kwon Kham SAI Giovanni VIGLIETTA Ryuhei UEHARA
We study a new reconfiguration problem inspired by classic mechanical puzzles: a colored token is placed on each vertex of a given graph; we are also given a set of distinguished cycles on the graph. We are tasked with rearranging the tokens from a given initial configuration to a final one by using cyclic shift operations along the distinguished cycles. We call this a cyclic shift puzzle. We first investigate a large class of graphs, which generalizes several classic cyclic shift puzzles, and we give a characterization of which final configurations can be reached from a given initial configuration. Our proofs are constructive, and yield efficient methods for shifting tokens to reach the desired configurations. On the other hand, when the goal is to find a shortest sequence of shifting operations, we show that the problem is NP-hard, even for puzzles with tokens of only two different colors.
Kunihiro WASA Katsuhisa YAMANAKA Hiroki ARIMURA
Given two feasible solutions A and B, a reconfiguration problem asks whether there exists a reconfiguration sequence (A0=A, A1,...,Aℓ=B) such that (i) A0,...,Aℓ are feasible solutions and (ii) we can obtain Ai from Ai-1 under the prescribed rule (the reconfiguration rule) for each i ∈ {1,...,ℓ}. In this paper, we address the reconfiguration problem for induced trees, where an induced tree is a connected and acyclic induced subgraph of an input graph. We consider the following two rules as the prescribed rules: Token Jumping: removing u from an induced tree and adding v to the tree, and Token Sliding: removing u from an induced tree and adding v adjacent to u to the tree, where u and v are vertices of an input graph. As the main results, we show that (I) the reconfiguration problemis PSPACE-complete even if the input graph is of bounded maximum degree, (II) the reconfiguration problem is W[1]-hard when parameterized by both the size of induced trees and the length of the reconfiguration sequence, and (III) there exists an FPT algorithm when the problem is parameterized by both the size of induced trees and the maximum degree of an input graph under Token Jumping and Token Sliding.
Takehiro ITO Kazuto KAWAMURA Xiao ZHOU
We study the problem of reconfiguring one list edge-coloring of a graph into another list edge-coloring by changing only one edge color assignment at a time, while at all times maintaining a list edge-coloring, given a list of allowed colors for each edge. Ito, Kami