matroids, graphs in surfaces, and the tutte polynomial - 2016 … · 2016-07-28 ·...
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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
Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
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
![Page 4: 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)](https://reader033.vdocuments.site/reader033/viewer/2022050612/5fb316225f64f04b7c7479dc/html5/thumbnails/4.jpg)
13
2 Tutte polynomial
Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
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
Unifying TopologicalTutte polynomials
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
4 Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
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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13
Tutte polynomial
4 Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
A Topological Tutte polynomial
I Deletion-contraction definition of the topologicalTutte polynomial:
![Page 8: 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)](https://reader033.vdocuments.site/reader033/viewer/2022050612/5fb316225f64f04b7c7479dc/html5/thumbnails/8.jpg)
13
Tutte polynomial
4 Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
A Topological Tutte polynomial
I Deletion-contraction definition of the topologicalTutte polynomial:
I No (full) recursive definition.
![Page 9: 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)](https://reader033.vdocuments.site/reader033/viewer/2022050612/5fb316225f64f04b7c7479dc/html5/thumbnails/9.jpg)
13
Tutte polynomial
4 Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
A Topological Tutte polynomial
I Deletion-contraction definition of the topologicalTutte polynomial:
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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13
Tutte polynomial
Topologicalextensions
5 A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
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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13
Tutte polynomial
Topologicalextensions
5 A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
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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13
Tutte polynomial
Topologicalextensions
6 A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
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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13
Tutte polynomial
Topologicalextensions
7 A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
(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
Topologicalextensions
8 A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
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
Topologicalextensions
9 A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
∆-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
Topologicalextensions
A matroidal setting
10 Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
‘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
Topologicalextensions
A matroidal setting
Matroidpolynomials
11 Graphicalanalogues
Unifying TopologicalTutte polynomials
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
Topologicalextensions
A matroidal setting
Matroidpolynomials
12 Graphicalanalogues
Unifying TopologicalTutte polynomials
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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13
Tutte polynomial
Topologicalextensions
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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13
Tutte polynomial
Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
13 Unifying TopologicalTutte polynomials
Thank You!
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13
Tutte polynomial
Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
13 Unifying TopologicalTutte polynomials