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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Vydáno v:Algorithmica Ročník 85; číslo 12; s. 3521 - 3571
Hlavní autoři: Binucci, Carla, Da Lozzo, Giordano, Di Giacomo, Emilio, Didimo, Walter, Mchedlidze, Tamara, Patrignani, Maurizio
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.12.2023
Springer Nature B.V
Témata:
Algorithm Analysis and Problem Complexity
Algorithms
Apexes
Combinatorial analysis
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Embedding
Exponential functions
Graph theory
Graphs
Mathematics of Computing
Questions
Theory of Computation
SPQR-trees
Branchwidth
st-graphs
Upward book embeddings
Treewidth
Sphere-cut decomposition
ISSN:0178-4617, 1432-0541
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Shrnutí: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