Matroids, graphs in surfaces, and the Tutte polynomial 2016 - - PowerPoint PPT Presentation

Matroids, graphs in surfaces, and the Tutte polynomial

2016 International Workshop on Structure in Graphs and Matroids Iain Moffatt and Ben Smith

Royal Holloway, University of London

Eindhoven, 29th July 2016

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T utte polynomial T

• pological

extensions A matroidal setting Matroid polynomials Graphical analogues Unifying T

• pological

T utte polynomials

Overview

◮ Introduce matroidal analogues of various notions of

embedded graphs.

◮ Introduce by applications to the theory of the T

utte polynomial:

• 1. Extensions of the T

utte polynomial to graphs in surfaces.

• 2. Incomplete aspects of the theory.
• 3. matroid model.
• 4. T
• pological graphs ↔ matroid models

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T utte polynomial T

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T utte polynomials

A review of the T utte polynomial

The Tutte polynomial, T(G; x, y)

Polynomial valued graph invariant, T : Graphs → Z[x, y].

◮ Importance due to applications / combinatorial info.

(colourings, flows, orientations, codes, Sandpile model, Potts & Ising models (statistical physics), QFT, Jones & homflypt polynomials (knot theory), ...)

Definition (deletion-contraction)

T(G; x, y) =          1 if G edgeless xT(G/e) if e a bridge yT(G\e) if e a loop T(G\e) + T(G/e)

• therwise

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T utte polynomial T

• pological

extensions A matroidal setting Matroid polynomials Graphical analogues Unifying T

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T utte polynomials

A review of the T utte polynomial

State sum formulation (T(G) is well-defined)

T(G) =

• A⊆E

(x − 1)r(G)−r(A)(y − 1)|A|−r(A) where r(A) = #verts. − #cpts. of (V, A) = rank of A .

◮ T is defined for matroids (e.g., r= rank function). ◮ T(C(G)) = T(G), where C(G) is cycle matroid ◮ Matroids often ‘complete’ graph results (e.g.

duality)

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T utte polynomials

Graphs in surfaces

◮ Plane graph - drawn on a sphere, edges don’t

meet, faces are disks.

◮ Embedded graph = graph in surface - drawn on

surface, edges don’t meet.

◮ Cellularly embedded graph - drawn on surface,

faces are disks.

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T

• pological

extensions A matroidal setting Matroid polynomials Graphical analogues Unifying T

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T utte polynomials

A T

• pological T

utte polynomial

The Bollobás-Riordan-Krushkal polynomial

K(G; x, y, a, b) :=

• A⊆E(G)

xr(G)−r(A)y|A|−r(A)aγ(A)bγ∗(Ac) γ(A) := Euler genus of nbhd. of subgraph of G on A γ∗(Ac) := Euler genus of nbhd. of subgraph of G∗ on Ac

◮ T(G; x, y) = K(G; x − 1, y − 1, 1, 1) ◮ G plane graph =

⇒ T(G; x, y) = K(G; x − 1, y − 1, a, b).

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T

• pological

extensions A matroidal setting Matroid polynomials Graphical analogues Unifying T

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T utte polynomials

A T

• pological T

utte polynomial

◮ Deletion-contraction definition of the topological

T utte polynomial:

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T

• pological

extensions A matroidal setting Matroid polynomials Graphical analogues Unifying T

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A T

• pological T

utte polynomial

◮ Deletion-contraction definition of the topological

T utte polynomial:

◮ No (full) recursive definition.

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T utte polynomial

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T

• pological

extensions A matroidal setting Matroid polynomials Graphical analogues Unifying T

• pological

T utte polynomials

A T

• pological T

utte polynomial

◮ Deletion-contraction definition of the topological

T utte polynomial:

◮ No (full) recursive definition. ◮ =

⇒ cell. embedded graphs are not the correct framework for the topological T utte polynomial!

◮ What is the correct framework?

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A matroidal setting Matroid polynomials Graphical analogues Unifying T

• pological

T utte polynomials

Look to matroids

◮ Why does deletion-contraction fail?

wants ribbon graph contraction wants graph contraction wants deletion as contraction in dual ¿ contract ?

◮ Exponents demand incompatible notions of

deletion and contraction....

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T utte polynomial T

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A matroidal setting Matroid polynomials Graphical analogues Unifying T

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Look to matroids

◮ Why does deletion-contraction fail?

wants ribbon graph contraction wants graph contraction wants deletion as contraction in dual ¿ contract ? Cycle matroid, C(G) Bond matroid, B(G*) Delta-matroid, D(G)

◮ Exponents demand incompatible notions of

deletion and contraction....

◮ ...but these are provided by various matroids.

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Delta-matroids

Symmetric Exchange Axiom (SEA): ∀X, Y ∈ F, if ∃u ∈ X△Y, then ∃v ∈ X△Y such that X△{u, v} ∈ F.

matroids (via bases)

M = (E, B)

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

⇒ |X| = |Y| Cycle matroid (trees) M(G) = (E, {{2}, {3}})

delta-matroids

M = (E, F)

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

⇒ |X| = |Y| ∆-matroid (quasi-trees) D(G) = (E, {{1, 2, 3}, {2}, {3}})

◮ Dmin = (E, {smallest sets}) a matroid ◮ Dmax = (E, {biggest sets}) a matroid ◮ D(G)min = C(G) ◮ D(G)max = B(G∗) = (C(G∗))∗

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(matroid, delta-matroid, matroid)

◮ Associate triple to embedded graph: ◮ Generally, consider triples

(M, D, N) of (matroid, delta-matroid, matroid)

◮ Deletion & contraction:

(M, D, N)\e := (M\e, D\e, N\e), (M, D, N)/e := (M/e, D/e, N/e)

◮ Important observation: different actions of deletion

contraction, (Dmin)/e = (D/e)min, (D\e)max = (Dmax\e). (So we have more than the delta-matroid.)

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Strong maps and matroid perspectives

◮ There is structure we are not seeing.

◮ Not all (graphic) triples can arise as minors of

(B(G∗), D(G), C(G)),

◮ e.g., 12 triples (M, D, N) on 1 element, only 5 arise. ◮

= ⇒ missing conditions.

Matroid perspectives

A matroid perspective, is a pair of matroids (M, N) over E such that

• 1. ⇐

⇒ every circuit of M is union of circuits of N

• 2. ⇐

⇒ every flat of N is a flat of M,

• 3. ⇐

⇒ rM(B) − rM(A) ≥ rN(B) − rN(A) when A ⊆ B ⊆ E

• 4. ⇐

⇒ M = H\A and N = H/A, for some H on E ⊔ A.

◮ Examples of matroid perspectives

◮ (B(G∗), C(G)) ◮ (C(G), C(H)) where H from G by identifying vertices ◮ (Dmax, Dmin) where D a delta-matroid

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∆-perspectives

∆-perspectives

An ∆-perspective is a triple (M, D, N) such that

• 1. M and N are matroids, and D is a

delta-matroid over the same set,

• 2. (M, Dmax) is a matroid perspective
• 3. (Dmin, N) is a matroid perspective

◮ Example: (B(G∗), D(G), C(G)) is a ∆-perspective.

Theorem

If (M, D, N) is an ∆-perspective, then so are (M, D, N)\e and (M, D, N)/e. (M, D, N) from cell. embed. graph its minors are.

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T utte polynomial T

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extensions A matroidal setting

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polynomials Graphical analogues Unifying T

• pological

T utte polynomials

‘T utte polynomial’ of perspectives

◮ There is a canonical way to construct ‘T

utte polynomials’ of objects (via Hopf algebras).

Definition: T utte polynomial of (M, D, N)

K(M, D, N) :=

• A⊆E

xr′(E)−r′(A)y|A|−r(A)aρ(A)−r′(A)br(A)−ρ(A), where ρ = 1

2(rmax + rmin). ◮ Theorems:

◮ Contains Bollobás-Riordan-Krushkal polynomial

K(G; x, y, a, b) = bγ(G)K((M, D, N); x, y, a2, b−2)

◮ K(M, D, N) has a 6 term deletion-contraction relation. ◮ duality formula, convolution formula, universality,...

◮ ∆-perspectives correct setting for topological T

utte polynomials.

◮ Results that should hold for BRK-polynomial but do

not, hold for the matroid version of the polynomial.

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T utte polynomial T

• pological

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analogues Unifying T

• pological

T utte polynomials

The graphical analogue

◮ Cellularly embedded graphs ↔ ∆-perspectives.

◮ Pseudo-surface = surface with

pinch points.

◮ Graph in pseudo surface - not

necessarily cell. embedded.

◮ Deletion and contraction defined in natural way:

delete contract

∆-persps. ↔ graphs in pseudo-surfaces

◮

→ (B(G∗), D(G), C(G)) =: P(G)

◮ P(G)/e = P(G/e), P(G)\e = P(G\e), (P(G))∗ = P(G∗) ◮ Bollobás-Riordan-Krushkal polynomial is not a

polynomial of cellularly embedded graphs.

◮ It is a polynomial of graphs in pseudo-surfaces.

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T utte polynomial T

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analogues Unifying T

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T utte polynomials

The graphical analogue of subobjects

◮ Natural sub-objects of (M, D, N).

◮ (M, D, N) ↔ graphs ◮ (M, D, N) ↔ cell. embed. in surfaces ◮ (M, D, N) ↔ cell. embed. in pseudo-surfaces ◮ (M, D, N) ↔ non-cell. embed. in surfaces ◮ (M, D, N) ↔ non-cell. embed. in pseudo-surfaces

◮ Concepts of minors, duals, etc. are compatible.

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• pological

T utte polynomials

Three T

• pological T

utte polynomials

◮ Various candidates for the topological T

utte polynomial in literature:

◮ M. Las Vergnas’ (1978), L(G; x, y, z) ◮ B. Bollobás and O. Riordan’s (2001/2), R(G; x, y, z) ◮ V. Kruskal’s (2011), K(G; x, y, a, b)

◮ Each corresponds to subobject ◮ =

⇒ each polynomial is a topological T utte polynomial but for a different notion of embedded graph.

◮ Challenge: use this to find new combinatorial

interpretations!

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T utte polynomial T

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extensions A matroidal setting Matroid polynomials Graphical analogues

13 Unifying T

• pological

T utte polynomials

Thank You!

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T utte polynomial T

• pological

extensions A matroidal setting Matroid polynomials Graphical analogues

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• pological

T utte polynomials