simple cycles, the case of splitting cycles · graphs homology simple cycles splitting cycles...
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![Page 1: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/1.jpg)
Simple cycles, the case of splittingcycles
Vincent DESPRÉ and Francis LAZARUS
gipsa-lab, G-SCOP, Labex Persyval, Grenoble, [email protected]
02/04/2015
![Page 2: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/2.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Surfaces
DefinitionA surface is a compact, connected 2-manifold withoutboundary.
GenusThe genus of a surface is its number of handles.
![Page 3: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/3.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Free homotopy
Two curves c1 and c2 are freely homotopic iff c1 can becontinously deform on c2.
We fix a base point x0 on the surface S. The correspondinghomotopy group is called π1(S).
![Page 4: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/4.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
![Page 5: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/5.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Simple cycle
A cycle on S is simple iff it has no double points.
![Page 6: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/6.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Embedded graphs
Embedding
An embedding of a graph G on a surface S is a properdrawing of G on S.
Cellular embedding
An embedding is a cellular embedding if the surface aftercutting along the graph is an union of open disks.
![Page 7: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/7.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Euler Characteristic
Definitionlet S be a surface of genus g and G a graph cellularlyembedded on S with v vertices, e edges and f faces, then:
χ(S) = v − e+ f = 2− 2g
χ(S) = 1− 2 + 1 = 2− 2 ∗ 1 = 0
![Page 8: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/8.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Homologous cycles
Two cycles on the graph C1 and C2 are homologous iff thereis a set a faces such that the boundary of the union of themare exactly C1 ∪ C2.
The homology group is called H1(S).
![Page 9: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/9.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Simple cycle
A cycle of G is simple iff it has no repeated vertex.
![Page 10: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/10.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Splitting cycle
3 kinds of cycles:õ Contractible and separating cycles (C1).õ Non-contractible and non-separating cycles (C2).õ Non-contractible and separating cycles also called
splitting cycles (C3).
C1
C3C2
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Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Different cuttings
![Page 12: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/12.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Examples
![Page 13: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/13.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Status of the problem
Complexity (Colin de Verdière et al. 2008)
Deciding if a combinatorial surface has a splitting cycle isNP-complete.
Conjecture (Barnette, 1982)
Every triangulation of a surface of genus at least 2 has asplitting cycle.
Conjecture (Mohar et Thomassen, 2001)
For all triangulation S of genus g and all h ∈ [[1, g − 1]], thereis a splitting cycle that separates S into two pieces ofgenera h and g − h.
![Page 14: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/14.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Results
Conjecture (Mohar et Thomassen, 2001)
For all triangulation S of genus g and all h ∈ [[1, g − 1]], thereis a splitting cycle that separates S into two pieces ofgenera h and g − h.
Counter-example
There is a triangulation of genus 20 without splitting cyclesfor h ∈ {5, · · · , 10}.
![Page 15: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/15.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Triangulations
DefinitionA triangulation of a surface S is a simplicial complex C anda homeomorphism between S and C.
This definition excludes:õ Loop edges.õ 2 edges with the same end points.õ 2 faces that share their 3 vertices.
![Page 16: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/16.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Irreducible triangulations
A triangulation is irreducible if none of its edges can becontracted.
For a fixed genus g, there is a finite number of irreducibletriangulations with bounded number of vertices (less than13g − 4 [Joret and Wood, 2010]) .
Consequence
The conjecture is decidable for fixed genus.
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Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Genus 2 irreducible triangulations
First implementation by Thom Sulanke.Genus 2:Number of triangulations: 396 785
• 3 4 5 6 7 8 Average
10 2 51 681 130 1 6.0911 2 58 2249 16138 7818 11 6.2112 25 1516 20507 72001 22877 121 6.0013 710 13004 50814 78059 16609 9 5.6114 8130 30555 12308 3328 205 1 4.2115 36794 1395 3 1 2 3.0416 661 3 3.0117 5 3.00
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Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Genus 6
We consider the 59 non-isomorphic embeddings of K12.
Average: 7.58Worst-case: 8
Average: 9.41Worst-case: 10
Average: 10.32Worst-case: 12 (Hamiltonian cycle!)
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Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Idea, case of K5
Kn has(n2
)edges.
Example of K5:
Number of edges:(52
)= 10
Number of faces: 3e = 2f ⇒ f = 23e =
203 /∈ N
![Page 20: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/20.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
General case
χ(S) = v − e+ f = n− n(n− 1)
2+
2
3· n(n− 1)
2= 2− 2g
g =(n− 3)(n− 4)
12
(n− 3)(n− 4) ≡ 0[12]⇔ n ≡ 0, 3, 4 or 7[12]
Theorem (Ringel and Youngs, ∼1970)
Kn can triangulate a surface if and only if n ≡ 0, 3, 4 or 7[12].
![Page 21: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/21.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
K7
![Page 22: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/22.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
K7
![Page 23: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/23.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Naïve algorithm
![Page 24: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/24.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Naïve algorithm
![Page 25: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/25.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Computation time
New implementation in C++. The data-structure used for thetriangulations is the flag representation (a.k.a.winged-edges).
n 12 15 16 19 31 43basic 2 s. 1 h. 12 h. ∼10 yearsfinal 10 s. 20 s. 25 s. 1 m. 25 m. 10 h.
This has been computed with an 8 cores computer with 16Go of RAM. It uses parallel computation.
![Page 26: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/26.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
An edge-coloring algorithm
![Page 27: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/27.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Results, case of K19
Genera Shortest splitting cycle #cycles1→ 19 10 20802→ 18 14 13743→ 17 18 2784→ 16 19 385→ 15 ⊥ 06→ 14 ⊥ 07→ 13 ⊥ 08→ 12 ⊥ 09→ 11 ⊥ 010→ 10 ⊥ 0
![Page 28: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/28.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
• 15 19 27 28 31 39 40 431 8 10 12 12 8 12 10 82 11 14 16 17 13 15 15 113 12 18 19 18 17 20 18 124 13 19 20 ⊥ 19 24 19 155 14 ⊥ 27 ⊥ 20 26 24 186 ⊥ ⊥ ⊥ 25 30 26 237 ⊥ ⊥ ⊥ 26 32 28 268 ⊥ ⊥ ⊥ 26 ⊥ 30 269 ⊥ ⊥ ⊥ ⊥ ⊥ 33 28
10 ⊥ ⊥ ⊥ ⊥ ⊥ 35 3411 ⊥ ⊥ ⊥ ⊥ 36 3412 ⊥ ⊥ ⊥ ⊥ 38 3513 ⊥ ⊥ ⊥ ⊥ 40 3514 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
K31: only 78 of the 130 980 splitting cycles are separatingthe surface into 2 pieces of genera 8 and 55.
![Page 29: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/29.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Known results
We talk about Kn with n = 12 · s+ i.
õ i = 0, 3, 4 and 7, there is an embedding (Ringel andYoungs 1972).
õ i = 3, 4 and 7, there are O(4s) non-isomorphicembeddings (Korzhik and Voss 2001).
õ n = 12 (NO), there are exactly 182 200 non-isomorphicembeddings (Ellingham and Stephen 2003).
õ n = 13 (NO), there are exactly 243 088 286non-isomorphic embeddings (Ellingham and Stephen2003).
õ i = 3 and 7 and for an infinite numbers of values of n,there are at least nc·n
22-colorable non-isomorphic
embeddings (Granell and Knor 2012).
![Page 30: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/30.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Example n ≡ 7
1
2
5
9 1
23
4
5
67
8
9
Rotation scheme of 0:
0 : (9, 7, 8, 3, 13, 15, 14, 11, 18, 4, 17, 10, 16, 5, 1, 12, 2, 6)
The other rotation scheme comes from the addition inZ/19Z:
1 : (10, 8, 9, 4, 14, 16, 15, 12, 0, 5, 18, 11, 17, 6, 2, 13, 3, 7)
![Page 31: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/31.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Face-width
Let G be a graph with a 2-cell embedding on a surface S.
Definition (Face-width)
The face-width of G is the minimum number of intersectionbetween any non-contractile cycle of S and the graph G.
TheoremLet f a face, k ∈ N∗ and Ef,k = {f ′, d(f, f ′) ≤ k}. Iffw(G) > 2k + 1 then there is a disk D such that:
Ef,k ⊂ D and ∂D ⊂ ∂Ef,k
![Page 32: Simple cycles, the case of splitting cycles · graphs Homology Simple cycles Splitting cycles Definition Different cuttings Preliminary results Complete graph embeddings Idea General](https://reader030.vdocuments.net/reader030/viewer/2022040617/5f1f3c28a335c62ed654e290/html5/thumbnails/32.jpg)
Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Face-width
TheoremLet f a face, k ∈ N∗ and Ef,k = {f ′, d(f, f ′) ≤ k}. Iffw(G) > 2k + 1 then there is a disk D such that:
Ef,k ⊂ D and ∂D ⊂ ∂Ef,k
f
k
f
k
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Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Conjectures
Conjecture (Zha, 1991)
Every combinatorial map of genus at least 2 and face-widthat least 3 has a splitting cycle.
Conjecture (Zha, 1991)
Every combinatorial map of genus g ≥ 2 and face-width atleast 3 has a splitting cycle that split the surface into onesurface of genus h and one of genus g − h, for all 0 < h < g.
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Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Optimality
A combinatorial map of genus 2 and face-width 2 withoutsplitting cycle:
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Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Results
In the orientable case:õ The conjecture is true for face-width at least 6 (Zha
and Zhao 1993).õ The conjecture is true for genus 2 and face-width
at least 4 (Ellingham and Zha 2003).In the non-orientable case:
õ The conjecture is true for face-width at least 5 (Zhaand Zhao 1993).
õ The conjecture is true for genus 2 (Robertson andThomas 1991).
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Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
Perspectives
õ Give a formal proof (without computer) that Ringel’sembedding of K19 do not have a splitting cycle thatseparates the surfaces into 2 pieces of genera 10.
õ Analyze the complexity of the edge-coloring algorithmin the case of complete graphs.
õ Prove the existence of splitting cycles of type 1 andg − 1 in every triangulation.
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Simple cycles
VincentDESPRE
SurfacesHomotopy
Simple cycles
EmbeddedgraphsHomology
Simple cycles
SplittingcyclesDefinition
Different cuttings
Preliminary results
CompletegraphembeddingsIdea
General case
Algorithmsand resultsNaïve algorithm
Heuristic
Results
Non-isomorphicembeddings
Another pointof viewFace-width
Perspectives
This isthe end