11,866 research outputs found
Making triangulations 4-connected using flips
We show that any combinatorial triangulation on n vertices can be transformed
into a 4-connected one using at most floor((3n - 9)/5) edge flips. We also give
an example of an infinite family of triangulations that requires this many
flips to be made 4-connected, showing that our bound is tight. In addition, for
n >= 19, we improve the upper bound on the number of flips required to
transform any 4-connected triangulation into the canonical triangulation (the
triangulation with two dominant vertices), matching the known lower bound of 2n
- 15. Our results imply a new upper bound on the diameter of the flip graph of
5.2n - 33.6, improving on the previous best known bound of 6n - 30.Comment: 22 pages, 8 figures. Accepted to CGTA special issue for CCCG 2011.
Conference version available at
http://2011.cccg.ca/PDFschedule/papers/paper34.pd
Once punctured disks, non-convex polygons, and pointihedra
We explore several families of flip-graphs, all related to polygons or
punctured polygons. In particular, we consider the topological flip-graphs of
once-punctured polygons which, in turn, contain all possible geometric
flip-graphs of polygons with a marked point as embedded sub-graphs. Our main
focus is on the geometric properties of these graphs and how they relate to one
another. In particular, we show that the embeddings between them are strongly
convex (or, said otherwise, totally geodesic). We also find bounds on the
diameters of these graphs, sometimes using the strongly convex embeddings.
Finally, we show how these graphs relate to different polytopes, namely type D
associahedra and a family of secondary polytopes which we call pointihedra.Comment: 24 pages, 6 figure
A History of Flips in Combinatorial Triangulations
Given two combinatorial triangulations, how many edge flips are necessary and
sufficient to convert one into the other? This question has occupied
researchers for over 75 years. We provide a comprehensive survey, including
full proofs, of the various attempts to answer it.Comment: Added a paragraph referencing earlier work in the vertex-labelled
setting that has implications for the unlabeled settin
Graph properties of graph associahedra
A graph associahedron is a simple polytope whose face lattice encodes the
nested structure of the connected subgraphs of a given graph. In this paper, we
study certain graph properties of the 1-skeleta of graph associahedra, such as
their diameter and their Hamiltonicity. Our results extend known results for
the classical associahedra (path associahedra) and permutahedra (complete graph
associahedra). We also discuss partial extensions to the family of nestohedra.Comment: 26 pages, 20 figures. Version 2: final version with minor correction
Flip-graph moduli spaces of filling surfaces
This paper is about the geometry of flip-graphs associated to triangulations
of surfaces. More precisely, we consider a topological surface with a
privileged boundary curve and study the spaces of its triangulations with n
vertices on the boundary curve. The surfaces we consider topologically fill
this boundary curve so we call them filling surfaces. The associated
flip-graphs are infinite whenever the mapping class group of the surface (the
group of self-homeomorphisms up to isotopy) is infinite, and we can obtain
moduli spaces of flip-graphs by considering the flip-graphs up to the action of
the mapping class group. This always results in finite graphs and we are
interested in their geometry.
Our main focus is on the diameter growth of these graphs as n increases. We
obtain general estimates that hold for all topological types of filling
surface. We find more precise estimates for certain families of filling
surfaces and obtain asymptotic growth results for several of them. In
particular, we find the exact diameter of modular flip-graphs when the filling
surface is a cylinder with a single vertex on the non-privileged boundary
curve.Comment: 52 pages, 29 figure
The geometry of flip graphs and mapping class groups
The space of topological decompositions into triangulations of a surface has
a natural graph structure where two triangulations share an edge if they are
related by a so-called flip. This space is a sort of combinatorial
Teichm\"uller space and is quasi-isometric to the underlying mapping class
group. We study this space in two main directions. We first show that strata
corresponding to triangulations containing a same multiarc are strongly convex
within the whole space and use this result to deduce properties about the
mapping class group. We then focus on the quotient of this space by the mapping
class group to obtain a type of combinatorial moduli space. In particular, we
are able to identity how the diameters of the resulting spaces grow in terms of
the complexity of the underlying surfaces.Comment: 46 pages, 23 figure
Topological Prismatoids and Small Simplicial Spheres of Large Diameter
We introduce topological prismatoids, a combinatorial abstraction of the
(geometric) prismatoids recently introduced by the second author to construct
counter-examples to the Hirsch conjecture. We show that the `strong -step
Theorem' that allows to construct such large-diameter polytopes from
`non--step' prismatoids still works at this combinatorial level. Then, using
metaheuristic methods on the flip graph, we construct four combinatorially
different non--step -dimensional topological prismatoids with
vertices. This implies the existence of -dimensional spheres with
vertices whose combinatorial diameter exceeds the Hirsch bound. These examples
are smaller that the previously known examples by Mani and Walkup in 1980 (
vertices, dimension ).
Our non-Hirsch spheres are shellable but we do not know whether they are
realizable as polytopes.Comment: 20 pages. Changes from v1 and v2: Reduced the part on shellability
and general improvement to accesibilit
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