Affiliation:
1. Technische Universität Berlin, Germany
2. Humboldt-Universität zu Berlin, Germany
Abstract
An out-tree
T
of a directed graph
D
is a rooted tree subgraph with all arcs directed outwards from the root. An out-branching is a spanning out-tree. By ℓ(
D
) and ℓ
s
(
D
), we denote the maximum number of leaves over all out-trees and out-branchings of
D
, respectively.
We give fixed parameter tractable algorithms for deciding whether ℓ
s
(
D
) ≥
k
and whether ℓ(
D
) ≥
k
for a digraph
D
on
n
vertices, both with time complexity 2
O
(
k
log
k
)
·
n
O
(1)
. This answers an open question whether the problem for out-branchings is in FPT, and improves on the previous complexity of 2
O
(
k
log
2
k
)
·
n
O
(1)
in the case of out-trees.
To obtain the complexity bound in the case of out-branchings, we prove that when all arcs of
D
are part of at least one out-branching, ℓ
s
(
D
) ≥ ℓ(
D
)/3. The second bound we prove in this article states that for strongly connected digraphs
D
with minimum in-degree 3, ℓ
s
(
D
) ≥ Θ(√
n
), where previously ℓ
s
(
D
) ≥ Θ(3√
n
) was the best known bound. This bound is tight, and also holds for the larger class of digraphs with minimum in-degree 3 in which every arc is part of at least one out-branching.
Funder
Deutsche Forschungsgemeinschaft
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
5 articles.
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