10,366 research outputs found
Orderly Spanning Trees with Applications
We introduce and study the {\em orderly spanning trees} of plane graphs. This
algorithmic tool generalizes {\em canonical orderings}, which exist only for
triconnected plane graphs. Although not every plane graph admits an orderly
spanning tree, we provide an algorithm to compute an {\em orderly pair} for any
connected planar graph , consisting of a plane graph of , and an
orderly spanning tree of . We also present several applications of orderly
spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem,
(2) the first area-optimal 2-visibility drawing of , and (3) the best known
encodings of with O(1)-time query support. All algorithms in this paper run
in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of
the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001),
Washington D.C., USA, January 7-9, 2001, pp. 506-51
Compact Floor-Planning via Orderly Spanning Trees
Floor-planning is a fundamental step in VLSI chip design. Based upon the
concept of orderly spanning trees, we present a simple O(n)-time algorithm to
construct a floor-plan for any n-node plane triangulation. In comparison with
previous floor-planning algorithms in the literature, our solution is not only
simpler in the algorithm itself, but also produces floor-plans which require
fewer module types. An equally important aspect of our new algorithm lies in
its ability to fit the floor-plan area in a rectangle of size (n-1)x(2n+1)/3.
Lower bounds on the worst-case area for floor-planning any plane triangulation
are also provided in the paper.Comment: 13 pages, 5 figures, An early version of this work was presented at
9th International Symposium on Graph Drawing (GD 2001), Vienna, Austria,
September 2001. Accepted to Journal of Algorithms, 200
On Overlapping Divergences
Using set-theoretic considerations, we show that the forest formula for
overlapping divergences comes from the Hopf algebra of rooted trees.Comment: 25p, latex, epsf for figures, final version, to appear in
Comm.Math.Phy
Schnyder woods for higher genus triangulated surfaces, with applications to encoding
Schnyder woods are a well-known combinatorial structure for plane
triangulations, which yields a decomposition into 3 spanning trees. We extend
here definitions and algorithms for Schnyder woods to closed orientable
surfaces of arbitrary genus. In particular, we describe a method to traverse a
triangulation of genus and compute a so-called -Schnyder wood on the
way. As an application, we give a procedure to encode a triangulation of genus
and vertices in bits. This matches the worst-case
encoding rate of Edgebreaker in positive genus. All the algorithms presented
here have execution time , hence are linear when the genus is fixed.Comment: 27 pages, to appear in a special issue of Discrete and Computational
Geometr
Transversal structures on triangulations: a combinatorial study and straight-line drawings
This article focuses on a combinatorial structure specific to triangulated
plane graphs with quadrangular outer face and no separating triangle, which are
called irreducible triangulations. The structure has been introduced by Xin He
under the name of regular edge-labelling and consists of two bipolar
orientations that are transversal. For this reason, the terminology used here
is that of transversal structures. The main results obtained in the article are
a bijection between irreducible triangulations and ternary trees, and a
straight-line drawing algorithm for irreducible triangulations. For a random
irreducible triangulation with vertices, the grid size of the drawing is
asymptotically with high probability up to an additive
error of \cO(\sqrt{n}). In contrast, the best previously known algorithm for
these triangulations only guarantees a grid size .Comment: 42 pages, the second version is shorter, focusing on the bijection
(with application to counting) and on the graph drawing algorithm. The title
has been slightly change
Orientation-Constrained Rectangular Layouts
We construct partitions of rectangles into smaller rectangles from an input
consisting of a planar dual graph of the layout together with restrictions on
the orientations of edges and junctions of the layout. Such an
orientation-constrained layout, if it exists, may be constructed in polynomial
time, and all orientation-constrained layouts may be listed in polynomial time
per layout.Comment: To appear at Algorithms and Data Structures Symposium, Banff, Canada,
August 2009. 12 pages, 5 figure
Schnyder decompositions for regular plane graphs and application to drawing
Schnyder woods are decompositions of simple triangulations into three
edge-disjoint spanning trees crossing each other in a specific way. In this
article, we define a generalization of Schnyder woods to -angulations (plane
graphs with faces of degree ) for all . A \emph{Schnyder
decomposition} is a set of spanning forests crossing each other in a
specific way, and such that each internal edge is part of exactly of the
spanning forests. We show that a Schnyder decomposition exists if and only if
the girth of the -angulation is . As in the case of Schnyder woods
(), there are alternative formulations in terms of orientations
("fractional" orientations when ) and in terms of corner-labellings.
Moreover, the set of Schnyder decompositions on a fixed -angulation of girth
is a distributive lattice. We also show that the structures dual to
Schnyder decompositions (on -regular plane graphs of mincut rooted at a
vertex ) are decompositions into spanning trees rooted at such
that each edge not incident to is used in opposite directions by two
trees. Additionally, for even values of , we show that a subclass of
Schnyder decompositions, which are called even, enjoy additional properties
that yield a reduced formulation; in the case d=4, these correspond to
well-studied structures on simple quadrangulations (2-orientations and
partitions into 2 spanning trees). In the case d=4, the dual of even Schnyder
decompositions yields (planar) orthogonal and straight-line drawing algorithms.
For a 4-regular plane graph of mincut 4 with vertices plus a marked
vertex , the vertices of are placed on a grid according to a permutation pattern, and in the orthogonal drawing
each of the edges of has exactly one bend. Embedding
also the marked vertex is doable at the cost of two additional rows and
columns and 8 additional bends for the 4 edges incident to . We propose a
further compaction step for the drawing algorithm and show that the obtained
grid-size is strongly concentrated around for a uniformly
random instance with vertices
K-6 minors in large 6-connected graphs
Jorgensen conjectured that every 6-connected graph with no K-6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs. (C) 2017 Published by Elsevier Inc
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