All Questions
Tagged with triangulations graph-theory
15 questions
6
votes
2
answers
404
views
Estimating shortest paths in planar drawings of graphs
Consider a drawing (in $\mathbb{R}^2$) of a planar graph. (The drawing is given, contrarily to the common setup in graph theory where we are seeking to build a drawing with specific properties.)
For ...
- 307
5
votes
1
answer
119
views
Sufficient condition for a Hamilton cycle $C$ in a planar triangulation $G$ s.t. every triangle in $G$ has an edge in $C$
Let $G$ be a $k$-connected planar triangulation ($k\geq 4$) and let $C$ be a Hamilton cycle of $G$. Then:
Which conditions would be sufficient to assure that every triangle of $G$ has at least one ...
2
votes
1
answer
138
views
Two ears polygon in a maximal planar hamiltonian graph
Given a maximal planar graph (+6vertices) without separating triangles. Then it can have many Hamilton cycles°. Such a cycle divides the graph into two triangulated polygons. Is it always possible to ...
- 35
8
votes
1
answer
618
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When is a triangulation of sphere two-colorable?
Let $T$ be a triangulation of sphere. We say that $T$ is $k$-colorable if the triangles of $T$ can be assigned with $k$ colors such that any two triangles with a common edge have different colors.
I ...
- 30.6k
3
votes
1
answer
136
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Surfaces generated by minimum-weight triangulations
The minimum-weight triangulation of simple polygons can be efficiently calculated in $O(n^3)$ time by dynamic programming, while it is $\text{NP-hard}$ for pointsets in general.
Looking at the ...
- 13.2k
3
votes
1
answer
142
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Uniqueness constraints for Delaunay triangulation
Commonly the assumption that is made on point sets that shall be Delaunay-triangulated is that no three are collinear and no four are cocircular.
Those assumptions are however too restrictive: if ...
- 13.2k
6
votes
2
answers
159
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Eberhard-type theorems for Fisk triangulations?
A triangulation of a surface is called a Fisk triangulation if the degree of all but two vertices is even, and these two exceptional vertices of odd degree are neighbors.
I would like to know what ...
- 19.1k
5
votes
6
answers
435
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Characterizing Convex Configurations of Quadrupels of Coplanar Points via Linear (In-)equalities between Distance Sums or Differences
Given 4 points $A$, $B$, $C$ and $D$ in general position in the euclidean plane, is it possible to determine from the 6 distances $AB$, $BC$, $CD$, $AD$, $AC$ and, $BD$ alone, whether every point is a ...
- 13.2k
0
votes
0
answers
37
views
Restrictions on crossing edges in Delaunay triangulations
what can be said about crossing edges in Delaunay triangulations, i.e. about pairs of edges that constitute to the heaviest perfect matching int the $K_4$ induced by the quadruplet of adjacent ...
- 13.2k
1
vote
3
answers
941
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How to do a clockwise ordering of a planar graph in order to define its faces?
I am currently making an algorithm for planar graphs that I need to triangulate so they become maximally planar (that is triangulated and planar) given only the lists of neighbors for each node : no ...
9
votes
1
answer
424
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Hamiltonian circuit
Let us consider a disk with one labelled point on the boundary and $n$ labelled points in the interior.
Let T be a triangulation of the whole disk with vertices on the labelled points such that T ...
- 828
1
vote
0
answers
254
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Interior and boundary vertices of weighted graphs
Xu He's article Rigidity of Infinite Disk Patterns and I have a problem with a statement he makes on page 7.
He considers weighted embedded planar graphs $G=(V, E)$ with weight function $\Theta: E \...
- 81
8
votes
1
answer
534
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coincidence between minimal triangulation numbers and chromatic numbers
A friend and I were reading up on the classification of compact surfaces when we realized that the minimal number of triangles in a triangulation of the sphere, torus, and projective plane are 4 (...
- 480
1
vote
1
answer
3k
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3D Delaunay Triangulation -> Euclidean Minimum Spanning Tree
I read that the Euclidean Minimum Spanning Tree (EMST) of a set of points is a subgraph of any Delaunay triangulation. Apparently the easiest/fastest way to obtain the EMST is to find the Deluanay ...
- 171
2
votes
1
answer
115
views
mean length of the non-crossing graphs on n points
My original question is rather vague so I'll start with a precise example and then indicate possible generalisations.
Given a n-tuple $x=(x_1,\dots,x_n)$ in, say, a square with side-length $1$ in the ...
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