7,859 research outputs found
Four Pages Are Indeed Necessary for Planar Graphs
An embedding of a graph in a book consists of a linear order of its vertices
along the spine of the book and of an assignment of its edges to the pages of
the book, so that no two edges on the same page cross. The book thickness of a
graph is the minimum number of pages over all its book embeddings. Accordingly,
the book thickness of a class of graphs is the maximum book thickness over all
its members. In this paper, we address a long-standing open problem regarding
the exact book thickness of the class of planar graphs, which previously was
known to be either three or four. We settle this problem by demonstrating
planar graphs that require four pages in any of their book embeddings, thus
establishing that the book thickness of the class of planar graphs is four
Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages
An embedding of a graph in a book, called book embedding, consists of a linear ordering of its vertices along the spine of the book and an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. For planar graphs, a fundamental result is due to Yannakakis, who proposed an algorithm to compute embeddings of planar graphs in books with four pages. Our main contribution is a technique that generalizes this result to a much wider family of nonplanar graphs, which is characterized by a biconnected skeleton of crossing-free edges whose faces have bounded degree. Notably, this family includes all 1-planar and all optimal 2-planar graphs as subgraphs. We prove that this family of graphs has bounded book thickness, and as a corollary, we obtain the first constant upper bound for the book thickness of optimal 2-planar graphs
A Linear Time Algorithm for Visualizing Knotted Structures in 3 Pages
We introduce simple codes and fast visualization tools for knotted structures in molecules and neural networks. Knots, links and more general knotted graphs are studied up to an ambient isotopy in Euclidean 3-space. A knotted graph can be represented by a plane diagram or by an abstract Gauss code. First we recognize in linear time if an abstract Gauss code represents an actual graph embedded in 3-space. Second we design a fast algorithm for drawing any knotted graph in the 3-page book, which is a union of 3 half-planes along their common boundary line. The running time of our drawing algorithm is linear in the length of a Gauss code of a given graph. Three-page embeddings provide simple linear codes of knotted graphs so that the isotopy problem for all graphs in 3-space completely reduces to a word problem in finitely presented semigroups
Book embeddings of Reeb graphs
Let be a simplicial complex with a piecewise linear function
. The Reeb graph is the quotient of , where we
collapse each connected component of to a single point. Let the
nodes of be all homologically critical points where any homology of
the corresponding component of the level set changes. Then we can
label every arc of with the Betti numbers
of the corresponding -dimensional
component of a level set. The homology labels give more information about the
original complex than the classical Reeb graph. We describe a canonical
embedding of a Reeb graph into a multi-page book (a star cross a line) and give
a unique linear code of this book embedding.Comment: 12 pages, 5 figures, more examples will be at http://kurlin.or
Optimal Acyclic Hamiltonian Path Completion for Outerplanar Triangulated st-Digraphs (with Application to Upward Topological Book Embeddings)
Given an embedded planar acyclic digraph G, we define the problem of "acyclic
hamiltonian path completion with crossing minimization (Acyclic-HPCCM)" to be
the problem of determining an hamiltonian path completion set of edges such
that, when these edges are embedded on G, they create the smallest possible
number of edge crossings and turn G to a hamiltonian digraph. Our results
include:
--We provide a characterization under which a triangulated st-digraph G is
hamiltonian.
--For an outerplanar triangulated st-digraph G, we define the st-polygon
decomposition of G and, based on its properties, we develop a linear-time
algorithm that solves the Acyclic-HPCCM problem with at most one crossing per
edge of G.
--For the class of st-planar digraphs, we establish an equivalence between
the Acyclic-HPCCM problem and the problem of determining an upward 2-page
topological book embedding with minimum number of spine crossings. We infer
(based on this equivalence) for the class of outerplanar triangulated
st-digraphs an upward topological 2-page book embedding with minimum number of
spine crossings and at most one spine crossing per edge.
To the best of our knowledge, it is the first time that edge-crossing
minimization is studied in conjunction with the acyclic hamiltonian completion
problem and the first time that an optimal algorithm with respect to spine
crossing minimization is presented for upward topological book embeddings
Symplectic fillings of Seifert fibered spaces
We give finiteness results and some classifications up to diffeomorphism of
minimal strong symplectic fillings of Seifert fibered spaces over S^2
satisfying certain conditions, with a fixed natural contact structure. In some
cases we can prove that all symplectic fillings are obtained by rational
blow-downs of a plumbing of spheres. In other cases, we produce new manifolds
with convex symplectic boundary, thus yielding new cut-and-paste operations on
symplectic manifolds containing certain configurations of symplectic spheres.Comment: 39 pages, 21 figures, v2 a few minor corrections and citations, v3
added clarifications in the proof of Lemma 2.8, plus some minor change
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