Linear-Time Algorithms for Tree Root Problems

Let T be a tree on a set V of nodes. The p-th power T (p) of T is the graph on V such that any two nodes u and w of V are adjacent in T (p) if and only if the distance of u and w in T is at most p. Given an n-node m-edge graph G and a positive integer p, the p-th tree root problem asks for a tree T, if any, such that G=T (p) . Given an n-node m-edge graph G, the tree root problem asks for a positive integer p and a tree T, if any, such that G=T (p) . Kearney and Corneil gave the best previously known algorithms for both problems. Their algorithm for the former (respectively, latter) problem runs in O(n (3)) (respectively, O(n (4))) time. In this paper, we give O(n+m)-time algorithms for both problems.