An Asymptotic Analysis of Labeled and Unlabeled k-Trees
In this paper we provide a systematic treatment of several shape parameters of (random) k -trees. Our research is motivated by many important algorithmic applications of k -trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k -trees are also a...
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| Published in: | Algorithmica Vol. 75; no. 4; pp. 579 - 605 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer US
01.08.2016
|
| Subjects: | |
| ISSN: | 0178-4617, 1432-0541 |
| Online Access: | Get full text |
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| Summary: | In this paper we provide a systematic treatment of several shape parameters of (random)
k
-trees. Our research is motivated by many important algorithmic applications of
k
-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand,
k
-trees are also a very interesting object from the combinatorial point of view. For both labeled and unlabeled
k
-trees, we prove that the number of
leaves
and more generally the number of
nodes
of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the
k
-tree. In particular we solve the
asymptotic counting problem
for unlabeled
k
-trees. By applying a proper singularity analysis of generating functions we show that the numbers
U
k
(
n
)
of unlabeled
k
-trees of size
n
are asymptotically given by
U
k
(
n
)
∼
c
k
n
-
5
/
2
ρ
k
-
n
, where
c
k
>
0
and
ρ
k
>
0
denotes the radius of convergence of the generating function
U
(
z
)
=
∑
n
≥
0
U
k
(
n
)
z
n
. |
|---|---|
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-015-0039-1 |