Cyclic Shift Problems on Graphs

We study a new reconfiguration problem inspired by classic mechanical puzzles: colored token is placed on each vertex of given graph; we are also set distinguished cycles the graph. tasked with rearranging tokens from initial configuration to final one using cyclic shift operations along cycles. first investigate large class graphs, which generalizes several puzzles, and give characterization configurations can be reached configuration. Our proofs constructive, yield efficient methods for shifting reach desired configurations. On other hand, when goal find shortest sequence operations, show that NP-hard, even puzzles only two different colors.