Algorithms for Square Roots of Graphs

The n-th power (n 1) of a graph G = (V; E), written G n , is deened to be the graph having V as its vertex set with two vertices u; v adjacent in G n if and only if there exists a path of length at most n between them. Similarly, graph H has an n-th root G if G n = H. For the case of n = 2, we say that G 2 is the square of G and G is the square root of G 2. Here we give a linear time algorithm for nding the tree square roots of a given graph and a linear time algorithm for nding the square roots of planar graphs. We also give a polynomial time algorithm for nding the square roots of subdivision graphs, which is equivalent to the problem of the inversion of total graphs. Further, we give a linear time algorithm for nding a Hamiltonian cycle in a cubic graph, and we prove the NP-completeness of nding the maximum cliques in powers of graphs and the chordality of powers of trees.