Parameterized Algorithms for Finding Square Roots

We show that the following two problems are fixed-parameter tractable with parameter k : testing whether a connected n -vertex graph with m edges has a square root with at most n-1+k edges and testing whether such a graph has a square root with at least m-k edges. Our first result implies that squares of graphs obtained from trees by adding at most k edges can be recognized in polynomial time for every fixed k≥ 0 ; previously this result was known only for k=0 . Our second result is equivalent to stating that deciding whether a graph can be modified into a square root of itself by at most k edge deletions is fixed-parameter tractable with parameter k .