Algorithms for Square Roots of Graphs

Abstract

The nth 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 nth rootG if $G^n = H$. For the case of $n = 2$, $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.