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.

如果存在一棵树T ，其叶子集为V ( G )，并且使得uv ∈ E ( G ) 当且仅当T中u和v之间的距离至多为k (且u ≠ v )。k的图类叶幂在计算生物学中有多种应用，但在过去的二十年里，识别它们仍然是一个具有挑战性的算法问题。最著名的结果是可以在多项式时间内识别 6 叶幂。在本文中，我们提出了一种算法，用于决定图G是否是时间O中的k叶幂（n ^f(k)对于仅依赖于k的某个函数f（但具有幂塔函数的增长率） 。

我们的技术基于以下事实：要么k叶幂具有较低最大度的对应树（在这种情况下很容易找到它），要么每个对应树都具有较大的最大度。在后一种情况下，树中的大度顶点意味着G具有可以从图中修剪的冗余子结构。除了解决长期存在的开放问题之外，我们希望这项工作中提出的结构结果能够在k叶幂和相关类别上带来进一步的结果。