  An st-shortest path, or st-path for short, in a graph G is a shortest
(induced) path from s to t in G. Two st-paths are said to be adjacent if they
differ on exactly one vertex. A reconfiguration sequence between two st-paths P
and Q is a sequence of adjacent st-paths starting from P and ending at Q.
Deciding whether there exists a reconfiguration sequence between two given
$st$-paths is known to be PSPACE-complete, even on restricted classes of graphs
such as graphs of bounded bandwidth (hence pathwidth). On the positive side,
and rather surprisingly, the problem is polynomial-time solvable on planar
graphs. In this paper, we study the parameterized complexity of the Shortest
Path Reconfiguration (SPR) problem. We show that SPR is W[1]-hard parameterized
by k + \ell, even when restricted to graphs of bounded (constant) degeneracy;
here k denotes the number of edges on an st-path, and \ell denotes the length
of a reconfiguration sequence from P to Q. We complement our hardness result by
establishing the fixed-parameter tractability of SPR parameterized by \ell and
restricted to nowhere-dense classes of graphs. Additionally, we establish
fixed-parameter tractability of SPR when parameterized by the treedepth, by the
cluster-deletion number, or by the modular-width of the input graph.
