On the Pagenumber of <i>k</i>-Trees

Vandenbussche, Jennifer; West, Douglas B.; Yu, Gexin

doi:10.1137/080714208

Cited by 11 publications

(10 citation statements)

References 9 publications

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“…This conjecture was refuted by Dujmović and Wood [6], who constructed a k-tree with book thickness k + 1 for all k ≥ 3. Vandenbussche et al [13] independently proved the same result. Therefore the maximum book thickness of a k-tree is k for k ≤ 2 and is k + 1 for k ≥ 3.…”

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confidence: 72%

“…This result is equivalent to saying that every k-tree that has a smooth degree-2 tree decomposition 2 of width k has a k-page book embedding. Vandenbussche et al [13] extended this result by showing that every k-tree that has a smooth degree-3 tree decomposition of width k has a k-page book embedding. Vandenbussche et al [13] then introduced the following natural definition.…”

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confidence: 95%

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On the book thickness of $k$-trees

Dujmović,

Wood

2009

Preprint

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Every k-tree has book thickness at most k + 1, and this bound is best possible for all k ≥ 3. Vandenbussche et al. (2009) proved that every k-tree that has a smooth degree-3 tree decomposition with width k has book thickness at most k. We prove this result is best possible for k ≥ 4, by constructing a k-tree with book thickness k + 1 that has a smooth degree-4 tree decomposition with width k. This solves an open problem of Vandenbussche et al. ( 2009) IntroductionConsider a drawing of a graph 1 G in which the vertices are represented by distinct points on a circle in the plane, and each edge is a chord of the circle between the corresponding points. Suppose that each edge is assigned one of k colours such that crossing edges receive distinct colours. This structure is called a k-page book embedding of G: one can also think of the vertices as being ordered along the spine of a book, and the edges that receive the same colour being drawn on a single page of the book without crossings. The book thickness of G, denoted by bt(G), is the minimum integer k for which there is a k-page book embedding of G. Book embeddings, first defined by Ollmann [9], are ubiquitous structures with a variety of applications; see [5] for a survey with over 50 references. A book embedding is also called a stack layout, and book thickness is also called stacknumber, pagenumber and fixed outerthickness.This paper focuses on the book thickness of k-trees.or G has a k-simplicial vertex v and G \ v is a k-tree. In the latter case, we say that G is obtained from G − v by adding v onto the k-clique N G (v).

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confidence: 72%

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confidence: 95%

On the book thickness of $k$-trees

Dujmović,

Wood

2009

Preprint

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“…That very graph satisfies pn (G) = 2, but we can augment it to a planar graph with local page number 3. [25] and pn(G) k + 1 if k 3 [13], and both bounds are best possible [10,30]. For the local and union page number we get a lower bound of k.…”

Section: Our Contributionmentioning

confidence: 99%

Local and Union Page Numbers

Merker

Ueckerdt

2019

Lecture Notes in Computer Science

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We introduce the novel concepts of local and union book embeddings, and, as the corresponding graph parameters, the local page number pn (G) and the union page number pn u (G). Both parameters are relaxations of the classical page number pn(G), and for every graph G we have pn (G) pn u (G) pn(G). While for pn(G) one minimizes the total number of pages in a book embedding of G, for pn (G) we instead minimize the number of pages incident to any one vertex, and for pn u (G) we instead minimize the size of a partition of G with each part being a vertex-disjoint union of crossing-free subgraphs. While pn (G) and pn u (G) are always within a multiplicative factor of 4, there is no bound on the classical page number pn(G) in terms of pn (G) or pn u (G). We show that local and union page numbers are closer related to the graph's density, while for the classical page number the graph's global structure can play a much more decisive role. We introduce tools to investigate local and union book embeddings in exemplary considerations of the class of all planar graphs and the class of graphs of tree-width k.As an incentive to pursue research in this new direction, we offer a list of intriguing open problems.

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“…In addition, upper bounds for the pagenumber of several graph classes are known, for example, complete bipartite graphs [7,17] and k-trees [5,8,21]. In particular, 1-page embeddable graphs and 2-page embeddable graphs are completely characterized as follows:…”

Section: Introductionmentioning

confidence: 99%

Book embedding of locally planar graphs on orientable surfaces

Nakamoto

Nozawa

2016

Discrete Mathematics

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a b s t r a c tA book embedding of a graph G is an embedding of vertices of G along the spine of a book, and edges of G on the pages so that no two edges on the same page intersect. Malitz (1994) proved that any graph on the orientable surface S g of genus g has a book embedding with O( √ g) pages. In this paper, we prove that every locally planar graph on S g (i.e., one with high representativity) has a book embedding with seven pages.

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On the Pagenumber of k-Trees

Cited by 11 publications

References 9 publications

On the book thickness of $k$-trees

On the book thickness of $k$-trees

Local and Union Page Numbers

Book embedding of locally planar graphs on orientable surfaces

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