The book thickness of a graph

Bernhart, Frank; Kainen, Paul C.

doi:10.1016/0095-8956(79)90021-2

Cited by 330 publications

(328 citation statements)

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“…Indeed, it was already known that a universal set of (exactly) n points supporting straightline drawings of planar graphs does not exist [3,4,9], while any set of n points can be universal if two bends per edge are allowed [8]. Also, notice that not all sets of points can be 1-bend universal: for example, if the points of S are collinear, exactly the family of sub-hamiltonian planar graphs has a 1-bend point-set embedding on S [1]. However, it is an open problem to determine whether any strictly convex point set is 1-bend universal.…”

Section: Resultsmentioning

confidence: 99%

“…Two planar graphs G 1 and G 2 with the same set of vertices are said to admit a simultaneous embedding without mapping if there exists a set of points in the plane that supports a point-set embedding of both G 1 and G 2 [2]. 1 It is not known whether any two planar graphs admit a simultaneous embedding without mapping such that all edges are straight-line segments. However, Brass et al [2] showed that any planar graph has a straight-line simultaneous embedding without mapping with any number of outerplanar graphs.…”

Section: Introductionmentioning

confidence: 99%

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Universal Sets of n Points for One-bend Drawings of Planar Graphs with n Vertices

Everett

Lazard

Liotta

et al. 2009

Discrete Comput Geom

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Universal Sets of n Points for One-bend Drawings of Planar Graphs with n Vertices

Everett

Lazard

Liotta

et al. 2009

Discrete Comput Geom

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“…The minimum number of pages required to construct such an embedding is known as book thickness or page number of a graph. An obvious upper bound on the page number of an n-vertex graph is n/2 , which is tight for complete graphs [3]. Book embeddings have a long history of research dating back to early seventies [19].…”

Section: Introductionmentioning

confidence: 99%

“…This is due to a result of Grigoriev and Bodlaender [9], and, independently of Kohrzik and Mohar [14], who proved that the problem of determining whether a graph is 1-planar is NP-hard (note that the problem remains NP-hard, even if the deletion of a single edge makes the graph planar [6]). In addition, we assume biconectivity, as it is known that the page number of a graph equals to the page number of its biconnected components [3]. …”

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confidence: 99%

The Book Thickness of 1-Planar Graphs is Constant

et al. 2016

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In a book embedding, the vertices of a graph are placed on the "spine" of a book and the edges are assigned to "pages", so that edges on the same page do not cross. In this paper, we prove that every 1-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant number of pages. To the best of our knowledge, the best non-trivial previous upper-bound is O( √ n), where n is the number of vertices of the graph. IntroductionA book embedding is a special type of a graph embedding, in which (i) the vertices of the graph are restricted to a line along the spine of a book, and, (ii) the edges on the pages of the book in such a way that edges residing on the same page do not cross. The minimum number of pages required to construct such an embedding is known as book thickness or page number of a graph. An obvious upper bound on the page number of an n-vertex graph is n/2 , which is tight for complete graphs [3]. Book embeddings have a long history of research dating back to early seventies [19]. Therefore, there is a rich body of literature (see, e.g., [4] and [20]). For the class of planar graphs, a central result is due to Yannakakis [23], who in the late eighties proved that planar graphs have book thickness at most four. It remains, however, unanswered whether the known bound of four is tight. Heath [10], for example, proves that all planar 3-trees are 3-page book embeddable. For more restricted subclasses of planar graphs, Bernhart and Kainen [3] show that the graphs with book thickness one are the outerplanar graphs, while the class of two-page embeddable graphs coincides with the class of subhamiltonian graphs (recall that subhamiltonian is a graph that is a subgraph of a planar Hamiltonian graph). Testing whether a graph is subhamiltonian is NP-complete [22]. However, several graph classes are known to be subhamiltonian (and therefore two-page book embeddable), e.g., 4-connected planar graphs [18], planar graphs without separating triangles [13], Halin graphs [7], planar graphs with maximum degree 3 or 4 [11,2].In this paper, we go a step beyond planar graphs. In particular, we consider 1-planar graphs and prove that their book thickness is constant. Recall that a graph is 1-planar, if it admits a drawing in which each edge is crossed at most once. To the best of our knowledge, the only (nontrivial) upper bound on the book thickness of 1-planar graphs on n vertices is O( √ n). This is * Electronic address: bekos@informatik. In the remainder of this paper, we will assume that a simple 1-planar drawing Γ(G) of the input 1-planar graph G is also specified as part of the input of the problem. This is due to a result of Grigoriev and Bodlaender [9], and, independently of Kohrzik and Mohar [14], who proved that the problem of determining whether a graph is 1-planar is NP-hard (note that the problem remains NP-hard, even if the deletion of a single edge makes the graph planar [6]). In addition, we assume biconectivity, as it...

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“…In general, the book embedding is hard: it is NP-complete to tell if a planar graph can be embedded in two pages [3]. It is known [2,3,12] that noncrossing graphs on a single page are exactly the outerplanar graphs [6]. However, no polylog parallel decision algorithm to tell if a graph is outerplanar is known.…”

Section: Introductionmentioning

confidence: 99%