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

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 $d$-angulations (plane graphs with faces of degree $d$) for all $d\geq 3$. A \emph{Schnyder decomposition} is a set of $d$ spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly $d-2$ of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the $d$-angulation is $d$. As in the case of Schnyder woods ($d=3$), there are alternative formulations in terms of orientations ("fractional" orientations when $d\geq 5$) and in terms of corner-labellings. Moreover, the set of Schnyder decompositions on a fixed $d$-angulation of girth $d$ is a distributive lattice. We also show that the structures dual to Schnyder decompositions (on $d$-regular plane graphs of mincut $d$ rooted at a vertex $v^*$) are decompositions into $d$ spanning trees rooted at $v^*$ such that each edge not incident to $v^*$ is used in opposite directions by two trees. Additionally, for even values of $d$, 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 $G$ of mincut 4 with $n$ vertices plus a marked vertex $v$, the vertices of $G\backslash v$ are placed on a $(n-1) \times (n-1)$ grid according to a permutation pattern, and in the orthogonal drawing each of the $2n-2$ edges of $G\backslash v$ has exactly one bend. Embedding also the marked vertex $v$ is doable at the cost of two additional rows and columns and 8 additional bends for the 4 edges incident to $v$. We propose a further compaction step for the drawing algorithm and show that the obtained grid-size is strongly concentrated around $25n/32\times 25n/32$ for a uniformly random instance with $n$ vertices

Some families of increasing planar maps

Stack-triangulations appear as natural objects when one wants to define some increasing families of triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with $2n$ faces under two different distributions. We show that the uniform distribution on this set of maps converges, for a topology of local convergence, to a distribution on the set of infinite maps. In the other hand, we show that rescaled by $n^{1/2}$, they converge for the Gromov-Hausdorff topology on metric spaces to the continuum random tree introduced by Aldous. Under a distribution induced by a natural random construction, the distance between random points rescaled by $(6/11)\log n$ converge to 1 in probability. We obtain similar asymptotic results for a family of increasing quadrangulations

4-labelings and grid embeddings of plane quadrangulations

We show that each quadrangulation on $n$ vertices has a closed rectangle of influence drawing on the $(n-2) \times (n-2)$ grid. Further, we present a simple algorithm to obtain a straight-line drawing of a quadrangulation on the $\Big\lceil\frac{n}{2}\Big\rceil \times \Big\lceil\frac{3n}{4}\Big\rceil$ 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 $4$-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

A topological classification of convex bodies

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale functions on S^2 by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in M_C (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph P_2 and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity-preserving local manipulation of the surface. Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist, this algorithm not only proves our claim but also generalizes the known classification scheme in [36]. Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. in [21], producing a hierarchy of increasingly coarse Morse-Smale complexes. We point out applications to pebble shapes.Comment: 25 pages, 10 figure

Which point sets admit a k-angulation?

For k >= 3, a k-angulation is a 2-connected plane graph in which every internal face is a k-gon. We say that a point set P admits a plane graph G if there is a straight-line drawing of G that maps V(G) onto P and has the same facial cycles and outer face as G. We investigate the conditions under which a point set P admits a k-angulation and find that, for sets containing at least 2k^2 points, the only obstructions are those that follow from Euler's formula.Comment: 13 pages, 7 figure