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.

10

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

10

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.

= =

10

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

10

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):

1

2

3

4 5

68

7

1

2

3

4 5

6

7

88

12

3

4

56

7

8

TheoremG represents link diagram ⇐⇒ no minor isomorphic to

, ,

10

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 .

10

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