We extend the well-studied concept of a graph power to that of a k-leaf power G of a tree T : G is formed by creating a node for each leaf in the tree and an edge between a pair of nodes if and only if the associated leaves are connected by a path of length at most k. By discovering hidden combinatorial structure of cliques and neighborhoods, we have developed polynomial-time algorithms that, for k = 3 and k = 4, identify whether or not a given graph G is a k-leaf power of a tree T , and if so… CONTINUE READING