Download - Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Deﬁnition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

Matroids, graphs in surfaces, and theTutte polynomial

2016 International Workshop on Structure in Graphs andMatroids

Iain Moffatt and Ben Smith

Royal Holloway, University of London

Eindhoven, 29th July 2016

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Tutte polynomial

Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

Overview

I Introduce matroidal analogues of various notions ofembedded graphs.

I Introduce by applications to the theory of the Tuttepolynomial:1. Extensions of the Tutte polynomial to graphs in

surfaces.2. Incomplete aspects of the theory.3. matroid model.4. Topological graphs ↔ matroid models

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2 Tutte polynomial


A matroidal setting

Matroidpolynomials

Graphicalanalogues


A review of the Tutte polynomial

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

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

I 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 edgelessxT(G/e) if e a bridgeyT(G\e) if e a loopT(G\e) + T(G/e) otherwise

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2 Tutte polynomial


A matroidal setting

Matroidpolynomials

Graphicalanalogues


A review of the Tutte 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 .

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

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Tutte polynomial

3 Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues


Graphs in surfaces

I Plane graph - drawn on a sphere, edges don’tmeet, faces are disks.

I Embedded graph = graph in surface - drawn onsurface, edges don’t meet.

I Cellularly embedded graph - drawn on surface,faces are disks.

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Tutte polynomial


A matroidal setting

Matroidpolynomials

Graphicalanalogues


A Topological Tutte polynomial

The Bollobás-Riordan-Krushkal polynomialK(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 Aγ∗(Ac) := Euler genus of nbhd. of subgraph of G∗ on Ac

I T(G;x,y) = K(G;x− 1,y − 1,1,1)

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

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Tutte polynomial


A matroidal setting

Matroidpolynomials

Graphicalanalogues



I Deletion-contraction definition of the topologicalTutte polynomial:

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Tutte polynomial


A matroidal setting

Matroidpolynomials

Graphicalanalogues




I No (full) recursive definition.

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Tutte polynomial


A matroidal setting

Matroidpolynomials

Graphicalanalogues




I No (full) recursive definition.I =⇒ cell. embedded graphs are not the correctframework for the topological Tutte polynomial!

I What is the correct framework?

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Tutte polynomial


5 A matroidal setting

Matroidpolynomials

Graphicalanalogues


Look to matroids

I Why does deletion-contraction fail?

wants ribbon graph contraction

wants graph contraction

wants deletion as contraction in dual

¿ contract ?

I Exponents demand incompatible notions ofdeletion and contraction....

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Tutte polynomial



Matroidpolynomials

Graphicalanalogues


Look to matroids

I 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)

I Exponents demand incompatible notions ofdeletion and contraction....

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

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Tutte polynomial



Matroidpolynomials

Graphicalanalogues


Delta-matroids

Symmetric Exchange Axiom (SEA): ∀X,Y ∈ F , if ∃u ∈ X4Y,then ∃v ∈ X4Y such that X4{u,v} ∈ F .

matroids (via bases)M = (E,B)

I B 6= ∅, subsets of EI B satisfies SEAI X,Y ∈ B =⇒ |X| = |Y|

Cycle matroid (trees)

M(G) = (E, {{2}, {3}})

delta-matroidsM = (E,F)

I F 6= ∅, subsets of EI F satisfies SEAI X,Y ∈ F =⇒ |X| = |Y|

∆-matroid (quasi-trees)

D(G) = (E, {{1,2,3}, {2}, {3}})

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

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Tutte polynomial



Matroidpolynomials

Graphicalanalogues


(matroid, delta-matroid, matroid)

I Associate triple to embedded graph:

I Generally, consider triples

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

I Deletion & contraction:

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

I Important observation: different actions of deletioncontraction,

(Dmin)/e 6= (D/e)min, (D\e)max 6= (Dmax\e).

(So we have more than the delta-matroid.)

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Tutte polynomial



Matroidpolynomials

Graphicalanalogues


Strong maps and matroid perspectives

I There is structure we are not seeing.I Not all (graphic) triples can arise as minors of

(B(G∗),D(G),C(G)),I e.g., 12 triples (M,D,N) on 1 element, only 5 arise.I =⇒ missing conditions.

Matroid perspectivesA matroid perspective, is a pair of matroids (M,N) overE such that1. ⇐⇒ every circuit of M is union of circuits of N2. ⇐⇒ every flat of N is a flat of M,3. ⇐⇒ rM(B)− rM(A) ≥ rN(B)− rN(A) when A ⊆ B ⊆ E4. ⇐⇒ M = H\A and N = H/A, for some H on E t A.

I Examples of matroid perspectivesI (B(G∗),C(G))I (C(G),C(H)) where H from G by identifying verticesI (Dmax,Dmin) where D a delta-matroid

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Tutte polynomial



Matroidpolynomials

Graphicalanalogues


∆-perspectives

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

1. M and N are matroids, and D is adelta-matroid over the same set,

2. (M,Dmax) is a matroid perspective3. (Dmin,N) is a matroid perspective

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

TheoremIf (M,D,N) is an ∆-perspective, then so are (M,D,N)\eand (M,D,N)/e.

(M,D,N) from cell. embed. graph ; its minors are.

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Tutte polynomial


A matroidal setting

10 Matroidpolynomials

Graphicalanalogues


‘Tutte polynomial’ of perspectives

I There is a canonical way to construct ‘Tuttepolynomials’ of objects (via Hopf algebras).

Definition: Tutte 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 = 12 (rmax + rmin).

I Theorems:I Contains Bollobás-Riordan-Krushkal polynomial

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

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

I ∆-perspectives correct setting for topological Tuttepolynomials.

I Results that should hold for BRK-polynomial but donot, hold for the matroid version of the polynomial.

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Tutte polynomial


A matroidal setting

Matroidpolynomials

11 Graphicalanalogues


The graphical analogue

I Cellularly embedded graphs 6↔ ∆-perspectives.

I Pseudo-surface = surface withpinch points.

I Graph in pseudo surface - notnecessarily cell. embedded.

I Deletion and contraction defined in natural way:delete contract

∆-persps. ↔ graphs in pseudo-surfaces

I 7→ (B(G∗),D(G),C(G)) =: P(G)

I P(G)/e = P(G/e), P(G)\e = P(G\e), (P(G))∗ = P(G∗)

I Bollobás-Riordan-Krushkal polynomial is not apolynomial of cellularly embedded graphs.

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

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Tutte polynomial


A matroidal setting

Matroidpolynomials

12 Graphicalanalogues


The graphical analogue of subobjects

I Natural sub-objects of (M,D,N).I (M,D,N) ↔ graphsI (M,D,N) ↔ cell. embed. in surfacesI (M,D,N) ↔ cell. embed. in pseudo-surfacesI (M,D,N) ↔ non-cell. embed. in surfacesI (M,D,N) ↔ non-cell. embed. in pseudo-surfaces

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

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Tutte polynomial


A matroidal setting

Matroidpolynomials

Graphicalanalogues

13 Unifying TopologicalTutte polynomials

Three Topological Tutte polynomials

I Various candidates for the topological Tuttepolynomial in literature:

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

I Each corresponds to subobject

I =⇒ each polynomial is a topological Tuttepolynomial but for a different notion of embeddedgraph.

I Challenge: use this to find new combinatorialinterpretations!

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Tutte polynomial


A matroidal setting

Matroidpolynomials

Graphicalanalogues


Thank You!

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Tutte polynomial


A matroidal setting

Matroidpolynomials

Graphicalanalogues


Download - Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Deﬁnition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

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