Upward Book Embeddability of st-Graphs: Complexity and Algorithms
A k - page upward book embedding ( k UBE) of a directed acyclic graph G is a book embeddings of G on k pages with the additional requirement that the vertices appear in a topological ordering along the spine of the book. The k UBE Testing problem, which asks whether a graph admits a k UBE, was int...
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Published in | Algorithmica Vol. 85; no. 12; pp. 3521 - 3571 |
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Main Authors | , , , , , |
Format | Journal Article |
Language | English |
Published |
New York
Springer US
01.12.2023
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | A
k
-
page upward book embedding
(
k
UBE) of a directed acyclic graph
G
is a book embeddings of
G
on
k
pages with the additional requirement that the vertices appear in a topological ordering along the spine of the book. The
k
UBE Testing
problem, which asks whether a graph admits a
k
UBE, was introduced in 1999 by Heath, Pemmaraju, and Trenk (SIAM J Comput 28(4), 1999). In a companion paper, Heath and Pemmaraju (SIAM J Comput 28(5), 1999) proved that the problem is linear-time solvable for
k
=
1
and NP-complete for
k
=
6
. Closing this gap has been a central question in algorithmic graph theory since then. In this paper, we make a major contribution towards a definitive answer to the above question by showing that
k
UBE Testing
is NP-complete for
k
≥
3
, even for
st
-graphs, i.e., acyclic directed graphs with a single source and a single sink. Indeed, our result, together with a recent work of Bekos et al. (Theor Comput Sci 946, 2023) that proves the NP-completeness of 2UBE for planar
st
-graphs, closes the question about the complexity of the
k
UBE problem for any
k
. Motivated by this hardness result, we then focus on the 2UBE
Testing
for planar
st
-graphs. On the algorithmic side, we present an
O
(
f
(
β
)
·
n
+
n
3
)
-time algorithm for 2UBE
Testing
, where
β
is the branchwidth of the input graph and
f
is a singly-exponential function on
β
. Since the treewidth and the branchwidth of a graph are within a constant factor from each other, this result immediately yields an FPT algorithm for
st
-graphs of bounded treewidth. Furthermore, we describe an
O
(
n
)-time algorithm to test whether a plane
st
-graph whose faces have a special structure admits a 2UBE that additionally preserves the plane embedding of the input
st
-graph. On the combinatorial side, we present two notable families of plane
st
-graphs that always admit an embedding-preserving
2
UBE. |
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ISSN: | 0178-4617 1432-0541 |
DOI: | 10.1007/s00453-023-01142-y |