A Linear Kernel for Finding Square Roots of Almost Planar Graphs
  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. 
Petr A. Golovach, Dieter Kratsch, Dani\"el Paulusma, and Anthony   Stewart
cs.DS cs.DM