Ribbon graphs and their minors Iain Moffatt Royal Holloway, - - PowerPoint PPT Presentation

Ribbon graphs and their minors

Iain Moffatt

Royal Holloway, University of London

British Combinatorial Conference, 9th July 2015

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Embedded graphs Ribbon graph minors Excluded minors Matroids

Graph minors

Graph minors

edge deletion vertex deletion edge contraction

H is a minor of G if it is

• btained by edge deletion,

edge contraction & vertex deletion.

Robertson-Seymour Theorem

◮ In any infinite collection of graphs, one graph is a

minor of another.

◮ Every minor-closed family of graphs is

characterised by a finite set of excluded minors.

◮ G can be embedded in R2 ⇐

⇒ no K5- or K3,3-minor.

◮ G can be embedded in RP2 ⇐

⇒ none of 35 minors.

◮ G can be embedded in surface Σ ⇐

⇒ none of finite list of minors.

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Embedded graphs Ribbon graph minors Excluded minors Matroids

Cellularly embedded graphs

G is cellularly embedded if it is drawn on a surface Σ such that

◮ edges don’t cross, ◮ faces are discs.

contraction deletion

not cell. embedded

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Embedded graphs

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Ribbon graph minors Excluded minors Matroids

Ribbon graphs

Ribbon graphs describe cellularly embedded graphs.

t a k e n e i g h b

• u

r h

• d

T a k e s p i n e d e l e t e f a c e s g l u e i n f a c e s

Ribbon graph

A “topological graph” with

◮ discs for vertices, ◮ ribbons for edges.

Considered up to homeomorphisms that preserve vertex-edge structure and cyclic order at vertices. = =

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Embedded graphs

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Ribbon graph minors Excluded minors Matroids

Ribbon graph minors

Edge and vertex deletion

e d g e d e l e t i

• n

vertex deletion

Edge contraction

G G/e non-loop n.-o. loop

• . loop

T

• contract

e = (u, v):

◮ attach a

disc to each ∂-cpt. of v ∪ e ∪ u

◮ remove

v ∪ e ∪ u

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Embedded graphs

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Ribbon graph minors Excluded minors Matroids

Ribbon graph minors

R.-S. theory for embedded graphs?

◮ Claim: the “correct” minors for embedded graphs. ◮ Conjecture: Every minor-closed family of ribbon

graphs is characterised by a finite set of excluded minors.

◮ But wait, is this not just a special case of

Robertson-Seymour?

◮ The two types of minor are incompatible.

◮ Contracting loops seems too hard. Can we just

delete loops like in the graph case?

◮ No, e.g.,

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Embedded graphs Ribbon graph minors

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Excluded minors Matroids

Excluded minor characterisations

Proposition

G is orientable ⇐ ⇒ no

• minor

Euler genus: γ(G) :=

• 2 × genus

if orientable genus if non-orientable

Theorem

G is of Euler genus ≤ n ⇐ ⇒ no minor in

◮ n odd: {G | γ(G) = n + 1, G = [1 vert., 1 ∂-cpt ]} ◮ n even: {G | (γ(G) = n + 1, G = [1 vert., 1 ∂-cpt ])

• r (γ(G) = n + 2, orient, G = [1 vert., 1 ∂-cpt ])}

Corollary

Orientable G is of genus ≤ n ⇐ ⇒ no minor in {G | (γ(G) = n + 2, orient G = [1 vert., 1 ∂-cpt ])}

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Embedded graphs Ribbon graph minors

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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 6 8 7 1 2 3 4 5 6 7 8 8 1 2 3 4 5 6 7 8

Theorem

G represents link diagram ⇐ ⇒ no minor isomorphic to , ,

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Embedded graphs Ribbon graph minors

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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}

Theorem

Partial dual of plane graph ⇐ ⇒ no minor isomorphic to , ,

Theorem

Partial dual of RP2 graph ⇐ ⇒ no minor isomorphic to , ,

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Embedded graphs Ribbon graph minors Excluded minors

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Matroids

A connection with matroids

Graph minors ← → Matroid minors (Robertson, Seymour) (Geelen, Gerards, Whittle)

matroids (via bases)

M = (E, B)

◮ B = ∅, subsets of E ◮ B satisfies SEA ◮ X, Y ∈ B =

⇒ |X| = |Y| Graphic matroid (trees)

1 3 2

MG = (E, {{2}, {3}})

delta-matroids

M = (E, F)

◮ F = ∅, subsets of E ◮ F satisfies SEA∗ ◮ X, Y ∈ F =

⇒ |X| = |Y| ∆-matroid (quasi-trees)

1 3 2

DG = (E, {{1, 2, 3}{2}, {3}})

∗ ∀X, Y ∈ F, u ∈ X△Y =

⇒ ∃v ∈ X△Y s.t. {u, v}△X ∈ F.

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Embedded graphs Ribbon graph minors Excluded minors

10 Matroids

Thank you!

◮ I. Moffatt, Ribbon graph minors and low-genus

partial duals, Annals Combin., to appear. arXiv:1502.00269

◮ I. Moffatt, Excluded minors and the graphs of knots,

• J. Graph Theory, to appear. arXiv:1311.2160

◮ C. Chun, I. Moffatt, S. Noble and R. Rueckriemen,

Matroids, Delta-matroids and Embedded Graphs,

• preprint. arXiv:1403.0920