  A graph H is a square root of a graph G if G can be obtained from H by the
addition of edges between any two vertices in H that are of distance 2 from
each other. The Square Root problem is that of deciding whether a given graph
admits a square root. We consider this problem for planar graphs in the context
of the "distance from triviality" framework. For an integer k, a planar+kv
graph (or k-apex graph) is a graph that can be made planar by the removal of at
most k vertices. We prove that a generalization of Square Root, in which some
edges are prescribed to be either in or out of any solution, has a kernel of
size O(k) for planar+kv graphs, when parameterized by k. Our result is based on
a new edge reduction rule which, as we shall also show, has a wider
applicability for the Square Root problem.
