6 research outputs found
Planar order on vertex poset
A planar order is a special linear extension of the edge poset (partially
ordered set) of a processive plane graph. The definition of a planar order
makes sense for any finite poset and is equivalent to the one of a conjugate
order. Here it was proved that there is a planar order on the vertex poset of a
processive planar graph naturally induced from the planar order of its edge
poset.Comment: 5 pages. Comments welcome. to appear in Journal of University of
Science and Technology of Chin
New bijective links on planar maps
This article describes new bijective links on planar maps, which are of incremental complexity and present original features. The first two bijections are correspondences on oriented planar maps. They can be considered as variations on the classical edge-poset construction for bipolar orientations on graphs, suitably adapted so as to operate only on the embeddings in a simple local way. In turn, yield two new bijections between families of (rooted) maps. (i) By identifying maps with specific constrained orientations, specialises to a bijection between 2-connected maps and irreducible triangulations; (ii) gives rise to a bijection between loopless maps and triangulations, observing that these decompose respectively into 2-connected maps and into irreducible triangulations in a parallel way
New bijective links on planar maps via orientations
This article presents new bijections on planar maps. At first a bijection is
established between bipolar orientations on planar maps and specific
"transversal structures" on triangulations of the 4-gon with no separating
3-cycle, which are called irreducible triangulations. This bijection
specializes to a bijection between rooted non-separable maps and rooted
irreducible triangulations. This yields in turn a bijection between rooted
loopless maps and rooted triangulations, based on the observation that loopless
maps and triangulations are decomposed in a similar way into components that
are respectively non-separable maps and irreducible triangulations. This gives
another bijective proof (after Wormald's construction published in 1980) of the
fact that rooted loopless maps with edges are equinumerous to rooted
triangulations with inner vertices.Comment: Extended and revised journal version of a conference paper with the
title "New bijective links on planar maps", which appeared in the Proceedings
of FPSAC'08, 23-27 June 2008, Vi\~na del Ma
On Schnyder's Theorm
The central topic of this thesis is Schnyder's Theorem. Schnyder's Theorem provides
a characterization of planar graphs in terms of their poset dimension, as follows: a graph
G is planar if and only if the dimension of the incidence poset of G is at most three. One
of the implications of the theorem is proved by giving an explicit mapping of the vertices
to R^2 that defines a straightline embedding of the graph. The other implication is proved
by introducing the concept of normal labelling. Normal labellings of plane triangulations
can be used to obtain a realizer of the incidence poset. We present an exposition of
Schnyder’s theorem with his original proof, using normal labellings. An alternate proof
of Schnyder’s Theorem is also presented. This alternate proof does not use normal
labellings, instead we use some structural properties of a realizer of the incidence poset
to deduce the result.
Some applications and a generalization of one implication of Schnyder’s Theorem
are also presented in this work. Normal labellings of plane triangulations can be used to
obtain a barycentric embedding of a plane triangulation, and they also induce a partition
of the edge set of a plane triangulation into edge disjoint trees. These two applications
of Schnyder’s Theorem and a third one, relating realizers of the incidence poset and
canonical orderings to obtain a compact drawing of a graph, are also presented. A
generalization, to abstract simplicial complexes, of one of the implications of Schnyder’s
Theorem was proved by Ossona de Mendez. This generalization is also presented in this
work. The concept of order labelling is also introduced and we show some similarities of
the order labelling and the normal labelling. Finally, we conclude this work by showing
the source code of some implementations done in Sage
Planarity and edge poset dimension
Different areas of discrete mathematics lead to intrinsically different characterizations of planar graphs. Planarity is expressed in terms of topology, combinatorics, algebra or search trees. More recently, Schnyder's work has related planarity to partial order theory. Acyclic orientations and associated edge partial orders lead to a new characterization of planar graphs, which also describes all the possible planar embeddings. We prove here that there is a bijection between bipolar plane digraphs and 2-dimensional N-free partial orders. We give also a characterization of planarity in terms of 2-colorability of a graph and provide a short proof of a previous result on planar lattices