31 research outputs found
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
Compatible 4-Holes in Point Sets
Counting interior-disjoint empty convex polygons in a point set is a typical
Erd\H{o}s-Szekeres-type problem. We study this problem for 4-gons. Let be a
set of points in the plane and in general position. A subset of ,
with four points, is called a -hole in if is in convex position and
its convex hull does not contain any point of in its interior. Two 4-holes
in are compatible if their interiors are disjoint. We show that
contains at least pairwise compatible 4-holes.
This improves the lower bound of which is implied by a
result of Sakai and Urrutia (2007).Comment: 17 page
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
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 2k² points, the only obstructions are those that follow from Euler’s formula
Ramified rectilinear polygons: coordinatization by dendrons
Simple rectilinear polygons (i.e. rectilinear polygons without holes or
cutpoints) can be regarded as finite rectangular cell complexes coordinatized
by two finite dendrons. The intrinsic -metric is thus inherited from the
product of the two finite dendrons via an isometric embedding. The rectangular
cell complexes that share this same embedding property are called ramified
rectilinear polygons. The links of vertices in these cell complexes may be
arbitrary bipartite graphs, in contrast to simple rectilinear polygons where
the links of points are either 4-cycles or paths of length at most 3. Ramified
rectilinear polygons are particular instances of rectangular complexes obtained
from cube-free median graphs, or equivalently simply connected rectangular
complexes with triangle-free links. The underlying graphs of finite ramified
rectilinear polygons can be recognized among graphs in linear time by a
Lexicographic Breadth-First-Search. Whereas the symmetry of a simple
rectilinear polygon is very restricted (with automorphism group being a
subgroup of the dihedral group ), ramified rectilinear polygons are
universal: every finite group is the automorphism group of some ramified
rectilinear polygon.Comment: 27 pages, 6 figure