Recognizing k -Leaf Powers in Polynomial Time, for Constant k

Abstract

A graph G is a k -leaf power if there exists a tree T whose leaf set is V ( G ), and such that uv ∈ E ( G ) if and only if the distance between u and v in T is at most k (and u ≠ v ). The graph classes of k -leaf powers have several applications in computational biology, but recognizing them has remained a challenging algorithmic problem for the past two decades. The best known result is that 6-leaf powers can be recognized in polynomial time. In this article, we present an algorithm that decides whether a graph G is a k -leaf power in time O ( n f(k) for some function f that depends only on k (but has the growth rate of a power tower function). Our techniques are based on the fact that either a k -leaf power has a corresponding tree of low maximum degree, in which case finding it is easy, or every corresponding tree has large maximum degree. In the latter case, large-degree vertices in the tree imply that G has redundant substructures which can be pruned from the graph. In addition to solving a long-standing open problem, it is our hope that the structural results presented in this work can lead to further results on k -leaf powers and related classes.