A graph G is a k-leaf power if there exists a tree T whose leaf set is V(G), and such that uv is an edge if and only if the distance between u and v in T is at most k. 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. In a recent paper presented at SODA22, it was shown that k-leaf powers can be recognized in polynomial time if k is fixed.

In this seminar, I will present the 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). More specifically, I will discuss how the difficult k-leaf power instances contain many cutsets that have the same neighborhood layering. I will then show that these similar cutsets are redundant and that removing one of them does not lose any information, which can be exploited for algorithmic purposes.