11 research outputs found
Binary Labelings for Plane Quadrangulations and their Relatives
Graphs and Algorithm
4-labelings and grid embeddings of plane quadrangulations
We show that each quadrangulation on vertices has a closed rectangle of influence drawing on the grid.
Further, we present a simple algorithm to obtain a straight-line drawing of a quadrangulation on the
grid.
This is not optimal but has the advantage over other existing algorithms that it is not needed to add edges to
the quadrangulation to make it -connected.
The algorithm is based on angle labeling and simple face counting in regions analogous to Schnyder's grid embedding for triangulation.
This extends previous results on book embeddings for quadrangulations from Felsner, Huemer, Kappes, and Orden (2008).
Our approach also yields a representation of a quadrangulation as a pair of rectangulations with a curious property
Combinatorial and Geometric Properties of Planar Laman Graphs
Laman graphs naturally arise in structural mechanics and rigidity theory.
Specifically, they characterize minimally rigid planar bar-and-joint systems
which are frequently needed in robotics, as well as in molecular chemistry and
polymer physics. We introduce three new combinatorial structures for planar
Laman graphs: angular structures, angle labelings, and edge labelings. The
latter two structures are related to Schnyder realizers for maximally planar
graphs. We prove that planar Laman graphs are exactly the class of graphs that
have an angular structure that is a tree, called angular tree, and that every
angular tree has a corresponding angle labeling and edge labeling.
Using a combination of these powerful combinatorial structures, we show that
every planar Laman graph has an L-contact representation, that is, planar Laman
graphs are contact graphs of axis-aligned L-shapes. Moreover, we show that
planar Laman graphs and their subgraphs are the only graphs that can be
represented this way.
We present efficient algorithms that compute, for every planar Laman graph G,
an angular tree, angle labeling, edge labeling, and finally an L-contact
representation of G. The overall running time is O(n^2), where n is the number
of vertices of G, and the L-contact representation is realized on the n x n
grid.Comment: 17 pages, 11 figures, SODA 201
4-labelings and grid embeddings of plane quadrangulations
AbstractA straight-line drawing of a planar graph G is a closed rectangle-of-influence drawing if for each edge uv, the closed axis-parallel rectangle with opposite corners u and v contains no other vertices. We show that each quadrangulation on n vertices has a closed rectangle-of-influence drawing on the (n−3)×(n−3) grid.The algorithm is based on angle labeling and simple face counting in regions. This answers the question of what would be a grid embedding of quadrangulations analogous to Schnyder’s classical algorithm for embedding triangulations and extends previous results on book embeddings for quadrangulations from Felsner, Huemer, Kappes, and Orden.A further compaction step yields a straight-line drawing of a quadrangulation on the (⌈n2⌉−1)×(⌈3n4⌉−1) grid. The advantage over other existing algorithms is that it is not necessary to add edges to the quadrangulation to make it 4-connected
Exploiting Air-Pressure to Map Floorplans on Point Sets
We prove a conjecture of Ackerman, Barequet and Pinter. Every floorplan with n segments can be embedded on every set of n points in generic position. The construction makes use of area universal floorplans also known as area universal rectangular layouts.
The notion of area used in our context depends on a nonuniform density function. We, therefore, have to generalize the theory of area universal floorplans to this situation. The method is then used to prove a result about accommodating points in floorplans that is slightly more general than the conjecture of Ackerman et al
Bijections for Baxter Families and Related Objects
The Baxter number can be written as . These
numbers have first appeared in the enumeration of so-called Baxter
permutations; is the number of Baxter permutations of size , and
is the number of Baxter permutations with descents and
rises. With a series of bijections we identify several families of
combinatorial objects counted by the numbers . Apart from Baxter
permutations, these include plane bipolar orientations with vertices and
faces, 2-orientations of planar quadrangulations with white and
black vertices, certain pairs of binary trees with left and
right leaves, and a family of triples of non-intersecting lattice paths. This
last family allows us to determine the value of as an
application of the lemma of Gessel and Viennot. The approach also allows us to
count certain other subfamilies, e.g., alternating Baxter permutations, objects
with symmetries and, via a bijection with a class of plan bipolar orientations
also Schnyder woods of triangulations, which are known to be in bijection with
3-orientations.Comment: 31 pages, 22 figures, submitted to JCT
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
Publicacions 2010. Campus del Baix Llobregat UPC
En aquest document s'han recopilat totes les publicacions realitzades pel professorat del Campus del Baix Llobregat de la UPC durant el perÃode comprés entre l'1 de Gener i el 31 d'Octubre de 2010.Preprin