Recognizing k-leaf powers in polynomial time, for constant k

TL;DR

This paper introduces an algorithm that recognizes k-leaf powers in polynomial time for fixed k, resolving a long-standing open problem by leveraging structural properties of the associated trees.

Contribution

It provides the first polynomial-time recognition algorithm for k-leaf powers for any fixed k, using novel structural insights about the trees representing these graphs.

Findings

Recognition algorithm runs in time O(n^{f(k)}) for some k-dependent function

Either the k-leaf power has a low-degree tree or can be simplified by pruning large degree vertices

Structural properties of k-leaf powers can be exploited for algorithmic recognition

Abstract

A graph $G$ is a $k$-leaf power if there exists a tree $T$ whose leaf set is $V(G)$, and such that $uv \in E(G)$ 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. The best known result is that $6$-leaf powers can be recognized in polynomial time. In this paper, 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 longstanding open problem, we hope that the structural results presented in this work can lead to further results on $k$-leaf powers.