Affiliation:
1. Université de Sherbrooke, Canada
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.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)