Algorithms for Square Roots of Graphs

The $n$th power ($n \geq 1$) of a graph $G = (V, E)$, written $G^n$, is defined 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 $G^2$. This paper presents a linear time algorithm for finding the tree square roots of a given graph and a linear time algorithm for finding the square roots of planar graphs. A polynomial time algorithm for finding the square roots of subdivision graphs, which is equivalent to the problem of the inversion of total graphs, is also presented. Further, the authors give a linear time algorithm for finding a Hamiltonian cycle in a cubic graph and prove the NP-completeness of finding the maximum cliques in powers of graphs and the chordality of powers of trees.