6 research outputs found
О рациональной укладке графов на плоскости
Секция 10. Теоретическая информатикаДоказано, что произвольное 3-дерево и произвольный связный геометриче-ский граф G = (V, E), такой, что для любой вершины v∈V имеет место deg(v) ≤ 4, причем существует вершина v0 ∈ V с deg(v0) < 4, можно небольшим шевелением
его вершин преобразовать в рациональный граф c вершинами, находящимися в общем положении
Constructing -clusters
A set of -lattice points in the plane, no three on a line and no four on a
circle, such that all pairwise distances and all coordinates are integral is
called an -cluster (in ). We determine the smallest existent
-cluster with respect to its diameter. Additionally we provide a toolbox of
algorithms which allowed us to computationally locate over 1000 different
-clusters, some of them having huge integer edge lengths. On the way, we
exhaustively determined all Heronian triangles with largest edge length up to
.Comment: 18 pages, 2 figures, 2 table
Distinct Distances in Graph Drawings
The \emph{distance-number} of a graph is the minimum number of distinct
edge-lengths over all straight-line drawings of in the plane. This
definition generalises many well-known concepts in combinatorial geometry. We
consider the distance-number of trees, graphs with no -minor, complete
bipartite graphs, complete graphs, and cartesian products. Our main results
concern the distance-number of graphs with bounded degree. We prove that
-vertex graphs with bounded maximum degree and bounded treewidth have
distance-number in . To conclude such a logarithmic upper
bound, both the degree and the treewidth need to be bounded. In particular, we
construct graphs with treewidth 2 and polynomial distance-number. Similarly, we
prove that there exist graphs with maximum degree 5 and arbitrarily large
distance-number. Moreover, as increases the existential lower bound on
the distance-number of -regular graphs tends to
STRAIGHT LINE EMBEDDINGS OF CUBIC PLANAR GRAPHS WITH INTEGER EDGE LENGTHS
Abstract. We prove that every simple cubic planar graph admits a planar embedding such that each edge is embedded as a straight line segment of integer length. 1
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