On reconfigurability of target sets

We study the problem of deciding reconfigurability of target sets of a graph. Given a graph G with vertex thresholds tau, consider a dynamic process in which vertex v becomes activated once at least tau(v) of its neighbors are activated. A vertex set S is called a target set if all vertices of G would be activated when initially activating vertices of S. In the TARGET SET RECONFIGURATION problem, given two target sets X and Y of the same size, we are required to determine whether X can be transformed into Y by repeatedly swapping one vertex in the current set with another vertex not in the current set preserving every intermediate set as a target set. In this paper, we investigate the complexity of TARGET SET RECONFIGURATION in restricted cases. On the hardness side, we prove that TARGET SET RECONFIGURATION is PSPACE-complete on bipartite planar graphs of degree 3 and 4 and of threshold 2, bipartite 3-regular graphs and planar 3-regular graphs of threshold 1 and 2, and split graphs, which is in contrast to the fact that a special case called VERTEX COVER RECONFIGURATION is in P for the last graph class. On the positive side, we present a polynomial-time algorithm for TARGET SET RECONFIGURATION on graphs of maximum degree 2 and trees. The latter result can be thought of as a generalization of that for VERTEX COVER RECONFIGURATION.