The Complexity of Induced Tree Reconfiguration Problems

A reconfiguration problem asks when we are given two feasible solutions A and B, whether there exists a reconfiguration sequence \((A_0 = A, A_1, \dots , A_\ell = B)\) such that (i) \(A_0, \dots , A_\ell \) are feasible solutions and (ii) we can obtain \(A_i\) from \(A_{i-1}\) under the prescribed rule (the reconfiguration rule) for each \(i = 1, \dots , \ell \). In this paper, we address the reconfiguration problem for induced trees, where an induced tree is a connected and acyclic induced graph of an input graph. This paper treats 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 in an input graph. As the main results, we show (I) the reconfiguration problem is PSPACE-complete, (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 parameterized by both the size of induced trees and the maximum degree of an input graph, under each of Token Jumping and Token Sliding.