Complexity of Finding Graph Roots with Girth Conditions

Graph G is the square of graph H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Given H it is easy to compute its square H (2), however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H (of girth 3). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if G=H (2) for some graph H of small girth. The main results are the following. There is a graph theoretical characterization for graphs that are squares of some graph of girth at least 7. A corollary is that if a graph G has a square root H of girth at least 7 then H is unique up to isomorphism. There is a polynomial time algorithm to recognize if G=H (2) for some graph H of girth at least 6. It is NP-complete to recognize if G=H (2) for some graph H of girth 4.