The Book Thickness of 1-Planar Graphs is Constant

Abstract

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...

Three-dimensional graph products with unbounded stack-number

“…The stack-number sn(G) of a graph G is the minimum s ∈ N 0 for which there exists an s-stack layout of G (also known as page-number , book-thickness or fixed outer-thickness). Stack layouts have been studied for planar graphs [9,17,40,61,62], graphs of given genus [31,41,47], treewidth [29,30,35,57], minor-closed graph classes [12], 1-planar graphs [1,7,8,16], and graphs with a given number of edges [48], amongst other examples.…”

Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

“…An algorithmic strategy to tackle the study of the planar slope number problem can be based on a peeling-into-levels approach. This approach has been successfully used to address the planar slope number problem for planar 3-trees [18], as well as to solve several other algorithmic problems on (near) planar graphs, including determining their pagenumber [6,5,16,26], computing their girth [8], and constructing radial drawings [12]. In the peelinginto-levels approach the vertices of a plane graph are partitioned into levels, based on their distance from the outer face.…”

Straight-line Drawings of 1-Planar Graphs

“…Geometric thickness specializes to book thickness if all vertices are placed in convex position. It is known that planar graphs have book thickness four [38], where the lower bound was proved just recently [39,40], and that the book thickness of 1-planar graphs is bounded by a constant [41].…”