On the Upward Book Thickness Problem: Combinatorial and Complexity Results

Bhore, Sujoy; Lozzo, Giordano Da; Montecchiani, Fabrizio; Nöllenburg, Martin

doi:10.1007/978-3-030-92931-2_18

Cited by 4 publications

(7 citation statements)

References 25 publications

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“…Conjecture 1 has received considerable attention, especially in recent years. Bhore, Da Lozzo, Montecchiani and Nöllenburg [11] confirm Conjecture 1 for several subclasses of outerplanar DAGs, including internally triangulated upward outerpaths or cacti. Subsequently, Nöllenburg and Pupyrev [41] confirm Conjecture 1 for single-source outerplanar DAGs, montone DAGs (to be defined in Section 2) and general outerpath DAGs.…”

Section: Related Worksupporting

confidence: 56%

“…In a Dagstuhl Report [6] Finally, let us mention that the decision problem of whether a given DAG admits a k-stack layout is known to be NP-complete for every k ≥ 2 [7,12,30]. However, there are FPT algorithms parameterized by the branchwidth [12] or the vertex cover number [11].…”

Section: Related Workmentioning

confidence: 99%

“…In fact, due to symmetry it is enough to consider a topological ordering ≺ with a 3 to the left of b 1 and to observe that in ≺ the edges a 1 b 1 , a 2 b 2 , a 3 b 3 form a 3-twist. Conjecture 1 concerning outerplanar DAGs, but also the more general question about directed acyclic 2-trees, have received increasing attention over the last years [16,11,41] (see Section 1.1 for more details), but it remained open to this day whether either of these classes contains DAGs of arbitrarily large stack number.…”

Section: Introductionmentioning

confidence: 99%

“…(Definitions are provided in Section 2.) This was known only for specific subclasses before, such as internally triangulated upward outerpaths [11]. As such, Theorem 2 provides one of the largest known classes of upward planar graphs with bounded stack number, while it is a famous open problem whether or not actually all upward planar graphs have bounded stack number [42].…”

Section: Introductionmentioning

confidence: 99%

“…Figure11 An outerplanar DAG with its construction tree and a 1-stack layout of the transitive subgraph below w…”

mentioning

confidence: 99%

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Directed Acyclic Outerplanar Graphs Have Constant Stack Number

Jungeblut¹,

Merker²,

Ueckerdt³

2022

Preprint

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The stack number of a directed acyclic graph G is the minimum k for which there is a topological ordering of G and a k-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We prove that the stack number of directed acyclic outerplanar graphs is bounded by a constant, which gives a positive answer to a conjecture by Heath, Pemmaraju and Trenk [SIAM J. Computing, 1999]. As an immediate consequence, this shows that all upward outerplanar graphs have constant stack number, answering a question by Bhore et al. [GD 2021] and thereby making significant progress towards the problem for general upward planar graphs originating from Nowakowski and Parker [Order, 1989]. As our main tool we develop the novel technique of directed H-partitions, which might be of independent interest.We complement the bounded stack number for directed acyclic outerplanar graphs by constructing a family of directed acyclic 2-trees that have unbounded stack number, thereby refuting a conjecture by Pupyrev [arXiv:2107.13658, 2021].

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Section: Related Worksupporting

confidence: 56%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Figure11 An outerplanar DAG with its construction tree and a 1-stack layout of the transitive subgraph below w…”

mentioning

confidence: 99%

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Directed Acyclic Outerplanar Graphs Have Constant Stack Number

Jungeblut¹,

Merker²,

Ueckerdt³

2022

Preprint

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Recognizing DAGs with Page-Number 2 Is NP-complete

Bekos

Lozzo

Frati

et al. 2023

Lecture Notes in Computer Science

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The page-number of a directed acyclic graph (a DAG, for short) is the minimum k for which the DAG has a topological order and a k-coloring of its edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological order. In 1999, Heath and Pemmaraju conjectured that the recognition of DAGs with page-number 2 is NP-complete and proved that recognizing DAGs with page-number 6 is NP-complete [SIAM J. Computing, 1999]. Binucci et al. recently strengthened this result by proving that recognizing DAGs with page-number k is NP-complete, for every k ≥ 3 [SoCG 2019]. In this paper, we finally resolve Heath and Pemmaraju's conjecture in the affirmative. In particular, our NP-completeness result holds even for stplanar graphs and planar posets.

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