5,061 research outputs found
Drawing planar graphs with prescribed face areas
This thesis deals with planar drawings of planar graphs such that each interior face has
a prescribed area.
Our work is divided into two main sections. The rst one deals with straight-line drawings
and the second one with orthogonal drawings.
For straight-line drawings, it was known that such drawings exist for all planar graphs
with maximum degree 3. We show here that such drawings exist for all planar partial 3-trees,
i.e., subgraphs of a triangulated planar graph obtained by repeatedly inserting a vertex in
one triangle and connecting it to all vertices of the triangle. Moreover, vertices have rational
coordinates if the face areas are rational, and we can bound the resolution.
For orthogonal drawings, we give an algorithm to draw triconnected planar graphs with
maximum degree 3. This algorithm produces a drawing with at most 8 bends per face and
4 bends per edge, which improves the previous known result of 34 bends per face. Both
vertices and bends have rational coordinates if the face areas are rational
Two-Page Book Embeddings of 4-Planar Graphs
Back in the Eighties, Heath showed that every 3-planar graph is
subhamiltonian and asked whether this result can be extended to a class of
graphs of degree greater than three. In this paper we affirmatively answer this
question for the class of 4-planar graphs. Our contribution consists of two
algorithms: The first one is limited to triconnected graphs, but runs in linear
time and uses existing methods for computing hamiltonian cycles in planar
graphs. The second one, which solves the general case of the problem, is a
quadratic-time algorithm based on the book-embedding viewpoint of the problem.Comment: 21 pages, 16 Figures. A shorter version is to appear at STACS 201
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
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