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Ribbon graphs and their minors
Iain Moffatt
Royal Holloway, University of London
British Combinatorial Conference, 9th July 2015
10
1 Embedded graphs
Ribbon graphminors
Excluded minors
Matroids
Graph minors
Graph minors
edge deletion
vertex deletion
edge contraction
H is a minor of G if it isobtained by edge deletion,edge contraction & vertexdeletion.
Robertson-Seymour TheoremI In any infinite collection of graphs, one graph is aminor of another.
I Every minor-closed family of graphs ischaracterised by a finite set of excluded minors.
I G can be embedded in R2 ⇐⇒ no K5- or K3,3-minor.I G can be embedded in RP2 ⇐⇒ none of 35 minors.I G can be embedded in surface Σ ⇐⇒ none offinite list of minors.
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2 Embedded graphs
Ribbon graphminors
Excluded minors
Matroids
Cellularly embedded graphs
G is cellularly embedded if it isdrawn on a surface Σ such that
I edges don’t cross,I faces are discs.
contraction
deletion
not cell.embedded
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Embedded graphs
3 Ribbon graphminors
Excluded minors
Matroids
Ribbon graphs
Ribbon graphs describe cellularly embedded graphs.
take neighbourhood
Take spine
delete faces
glue in faces
Ribbon graph
A “topological graph”with
I discs for vertices,I ribbons for edges.
Considered up to homeomorphisms that preservevertex-edge structure and cyclic order at vertices.
= =
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Embedded graphs
4 Ribbon graphminors
Excluded minors
Matroids
Ribbon graph minors
Edge and vertex deletion
edge deletion
vertex deletio
n
Edge contractionG G/e
non-loop
n.-o. loop
o. loop
To contracte = (u,v):
I attach adisc to each∂-cpt. ofv ∪ e ∪ u
I removev ∪ e ∪ u
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Embedded graphs
5 Ribbon graphminors
Excluded minors
Matroids
Ribbon graph minors
R.-S. theory for embedded graphs?I Claim: the “correct” minors for embedded graphs.I Conjecture: Every minor-closed family of ribbongraphs is characterised by a finite set of excludedminors.
I But wait, is this not just a special case ofRobertson-Seymour?
I The two types of minor are incompatible.I Contracting loops seems too hard. Can we justdelete loops like in the graph case?
I No, e.g.,
10
Embedded graphs
Ribbon graphminors
6 Excluded minors
Matroids
Excluded minor characterisations
Proposition
G is orientable ⇐⇒ no -minor
Euler genus: γ(G) :=
{2× genus if orientablegenus if non-orientable
TheoremG is of Euler genus ≤ n ⇐⇒ no minor in
I n odd: {G | γ(G) = n + 1,G =⋃
[1 vert., 1 ∂-cpt ]}I n even: {G | (γ(G) = n + 1,G =
⋃[1 vert., 1 ∂-cpt ])
or (γ(G) = n + 2, orient, G =⋃
[1 vert., 1 ∂-cpt ])}
CorollaryOrientable G is of genus ≤ n ⇐⇒ no minor in{G | (γ(G) = n + 2, orient G =
⋃[1 vert., 1 ∂-cpt ])}
10
Embedded graphs
Ribbon graphminors
7 Excluded minors
Matroids
Excluded minor characterisations
Knots & links can be represented by ribbon graphs(Dasbach, Futer, Kalfagianni, Lin, Stoltzfus, ’05):
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2
3
4 5
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1
2
3
4 5
6
7
88
12
3
4
56
7
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TheoremG represents link diagram ⇐⇒ no minor isomorphic to
, ,
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Embedded graphs
Ribbon graphminors
8 Excluded minors
Matroids
Excluded minor characterisations
Partial dual – form the dual w.r.t. only some edges.
G G{1} G{1,2} G∗ = G{1,2,3}
TheoremPartial dual of plane graph ⇐⇒ no minor isomorphic to
, ,
TheoremPartial dual of RP2 graph ⇐⇒ no minor isomorphic to
, ,
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Embedded graphs
Ribbon graphminors
Excluded minors
9 Matroids
A connection with matroids
Graph minors ←→ Matroid minors(Robertson, Seymour) (Geelen, Gerards, Whittle)
matroids (via bases)M = (E,B)
I B 6= ∅, subsets of EI B satisfies SEAI X,Y ∈ B =⇒ |X| = |Y|
Graphic matroid (trees)1
3
2
MG = (E, {{2}, {3}})
delta-matroidsM = (E,F)
I F 6= ∅, subsets of EI F satisfies SEA∗I X,Y ∈ F =⇒ |X| = |Y|
∆-matroid (quasi-trees)1
3
2
DG = (E, {{1,2,3}{2}, {3}})
∗ ∀X,Y ∈ F , u ∈ X4Y =⇒ ∃v ∈ X4Y s.t. {u,v}4X ∈ F .
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Embedded graphs
Ribbon graphminors
Excluded minors
10 MatroidsThank you!
I I. Moffatt, Ribbon graph minors and low-genuspartial duals, Annals Combin., to appear.arXiv:1502.00269
I I. Moffatt, Excluded minors and the graphs of knots,J. Graph Theory, to appear. arXiv:1311.2160
I C. Chun, I. Moffatt, S. Noble and R. Rueckriemen,Matroids, Delta-matroids and Embedded Graphs,preprint. arXiv:1403.0920
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