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Minors and Tutte invariants for alternating...
Transcript of Minors and Tutte invariants for alternating...
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Minors and Tutte invariantsfor alternating dimaps
Graham Farr
Clayton School of ITMonash University
Work done partly at: Isaac Newton Institute for Mathematical Sciences (Combinatorics and StatisticalMechanics Programme), Cambridge, 2008; University of Melbourne (sabbatical), 2011; and Queen Mary,
University of London, 2011.
20 March 2014
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Contraction and Deletion
G
e
u v
G \ e
u v
G/e
u = v
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Minors
H is a minor of G if it can be obtained from G by some sequenceof deletions and/or contractions.
The order doesn’t matter. Deletion and contraction commute:
G/e/f = G/f /e
G \ e \ f = G \ f \ e
G/e \ f = G \ f /e
Importance of minors:I excluded minor characterisations
I planar graphs (Kuratowski, 1930; Wagner, 1937)I graphs, among matroids (Tutte, PhD thesis, 1948)I Robertson-Seymour Theorem (1985–2004)
I countingI Tutte-Whitney polynomial family
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Minors
H is a minor of G if it can be obtained from G by some sequenceof deletions and/or contractions.
The order doesn’t matter. Deletion and contraction commute:
G/e/f = G/f /e
G \ e \ f = G \ f \ e
G/e \ f = G \ f /e
Importance of minors:I excluded minor characterisations
I planar graphs (Kuratowski, 1930; Wagner, 1937)I graphs, among matroids (Tutte, PhD thesis, 1948)I Robertson-Seymour Theorem (1985–2004)
I countingI Tutte-Whitney polynomial family
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Duality and minors
Classical duality for embedded graphs:
G ←→ G ∗
vertices ←→ faces
contraction ←→ deletion
(G/e)∗ = G ∗ \ e
(G \ e)∗ = G ∗/e
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Duality and minors
Classical duality for embedded graphs:
G ←→ G ∗
vertices ←→ faces
contraction ←→ deletion
(G/e)∗ = G ∗ \ e
(G \ e)∗ = G ∗/e
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Duality and minors
G
G/e
G \ e
G ∗
G ∗/e
G ∗ \ e
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Duality and minors
G
G/e
G \ e
G ∗
G ∗/e
G ∗ \ e
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Duality and minors
G
G/e
G \ e
G ∗
G ∗/e
G ∗ \ e
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Duality and minors
G
G/e
G \ e
G ∗
G ∗/e
G ∗ \ e
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Loops and coloops
loop coloop = bridge = isthmus
duality
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Loops and coloops
loop coloop = bridge = isthmus
duality
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History
H. E. Dudeney,Puzzling Times atSolvamhall Castle:Lady Isabel’s Casket,London Magazine7 (42) (Jan 1902) 584
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History
London Magazine8 (43) (Feb 1902) 56
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History
First published by Heinemann, London, 1907. Above is from 4th edn, Nelson, 1932.
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History
Duke Math. J. 7 (1940) 312–340.
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History
from a design for a proposed memorial to Tutte in Newmarket, UK.
https://www.facebook.com/billtutte
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History
Proc. Cambridge Philos. Soc. 44 (1948) 463–482.
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triad of alternating dimapsbicubic map
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triad of alternating dimapsbicubic map
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triad of alternating dimapsbicubic map
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triad of alternating dimapsbicubic map
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triad of alternating dimapsbicubic map
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triad of alternating dimapsbicubic map
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triad of alternating dimapsbicubic map
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triad of alternating dimapsbicubic map
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triad of alternating dimapsbicubic map
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triad of alternating dimaps
bicubic map
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triad of alternating dimaps
bicubic map
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triad of alternating dimapsbicubic map
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triad of alternating dimapsbicubic map
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triad of alternating dimaps
bicubic map
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triad of alternating dimaps
bicubic map
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Alternating dimapsAlternating dimap (Tutte, 1948):
I directed graph without isolated vertices,I 2-cell embedded in a disjoint union of orientable 2-manifolds,I each vertex has even degree,I ∀v : edges incident with v are directed alternately into, and
out of, v (as you go around v).
So vertices look like this:
Genus γ(G ) of an alternating dimap G :
V − E + F = 2(k(G )− γ(G ))
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Alternating dimapsAlternating dimap (Tutte, 1948):
I directed graph without isolated vertices,I 2-cell embedded in a disjoint union of orientable 2-manifolds,I each vertex has even degree,I ∀v : edges incident with v are directed alternately into, and
out of, v (as you go around v).
So vertices look like this:
Genus γ(G ) of an alternating dimap G :
V − E + F = 2(k(G )− γ(G ))
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Alternating dimapsAlternating dimap (Tutte, 1948):
I directed graph without isolated vertices,I 2-cell embedded in a disjoint union of orientable 2-manifolds,I each vertex has even degree,I ∀v : edges incident with v are directed alternately into, and
out of, v (as you go around v).
So vertices look like this:
Genus γ(G ) of an alternating dimap G :
V − E + F = 2(k(G )− γ(G ))
![Page 54: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/54.jpg)
Alternating dimaps
Three special partitions of E (G ):• clockwise faces
σc
• anticlockwise faces
σa
• in-stars
σi
(An in-star is the set of all edges going into some vertex.)
Each defines a permutation of E (G ). These permutations satisfy
σiσcσa = 1
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Alternating dimaps
Three special partitions of E (G ):• clockwise faces
σc
• anticlockwise faces
σa
• in-stars
σi
(An in-star is the set of all edges going into some vertex.)Each defines a permutation of E (G ).
These permutations satisfy
σiσcσa = 1
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Alternating dimaps
Three special partitions of E (G ):• clockwise faces σc• anticlockwise faces σa• in-stars σi
(An in-star is the set of all edges going into some vertex.)Each defines a permutation of E (G ).
These permutations satisfy
σiσcσa = 1
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Alternating dimaps
Three special partitions of E (G ):• clockwise faces σc• anticlockwise faces σa• in-stars σi
(An in-star is the set of all edges going into some vertex.)Each defines a permutation of E (G ). These permutations satisfy
σiσcσa = 1
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Triality (Trinity)
Construction of trial map:
clockwise faces −→ vertices −→ anticlockwise faces −→ clockwise faces
(σi , σc , σa) 7→ (σc , σa, σi )
u
e f
vC1 vC2
eω
![Page 59: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/59.jpg)
Triality (Trinity)
Construction of trial map:
clockwise faces −→ vertices −→ anticlockwise faces −→ clockwise faces
(σi , σc , σa) 7→ (σc , σa, σi )
u
e f
vC1 vC2
eω
![Page 60: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/60.jpg)
Triality (Trinity)
Construction of trial map:
clockwise faces −→ vertices −→ anticlockwise faces −→ clockwise faces
(σi , σc , σa) 7→ (σc , σa, σi )
u
e f
vC1 vC2
eω
![Page 61: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/61.jpg)
Triality (Trinity)
Construction of trial map:
clockwise faces −→ vertices −→ anticlockwise faces −→ clockwise faces
(σi , σc , σa) 7→ (σc , σa, σi )
u
e f
vC1 vC2
eω
![Page 62: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/62.jpg)
Minor operations
G
u
v
e
w1 w2
![Page 63: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/63.jpg)
Minor operations
G [1]e u = v
w1 w2
![Page 64: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/64.jpg)
Minor operations
G
u
v
e
w1 w2
![Page 65: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/65.jpg)
Minor operations
G [ω]e
u
v
w1 w2
![Page 66: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/66.jpg)
Minor operations
G
u
v
e
w1 w2
![Page 67: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/67.jpg)
Minor operations
G [ω2]e
u
v
w1 w2
![Page 68: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/68.jpg)
Minor operations
G
u
v
e
w1 w2
eω
![Page 69: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/69.jpg)
Minor operations
G
u
v
e
w1 w2
eω
![Page 70: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/70.jpg)
Minor operations
(
G [1]e
)ω = Gω[ω2]eω
u = v
w1 w2
![Page 71: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/71.jpg)
Minor operations
(G [1]e)ω = Gω[ω2]eω u = v
w1 w2
![Page 72: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/72.jpg)
Minor operations
Gω[1]eω = (G [ω]e)ω,
Gω[ω]eω = (G [ω2]e)ω,
Gω[ω2]eω = (G [1]e)ω,
Gω2[1]eω
2= (G [ω2]e)ω
2,
Gω2[ω]eω
2= (G [1]e)ω
2,
Gω2[ω2]eω
2= (G [ω]e)ω
2.
TheoremIf e ∈ E (G ) and µ, ν ∈ {1, ω, ω2} then
Gµ[ν]eω = (G [µν]e)µ.
Same pattern as established for other generalised minor operations(GF, 2008/2013. . . ).
![Page 73: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/73.jpg)
Minor operations
G
G [1]e
G [ω]e
G [ω2]e
Gω
Gω[1]e
Gω[ω]e
Gω[ω2]e
Gω2
Gω2[1]e
Gω2[ω]e
Gω2[ω2]e
![Page 74: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/74.jpg)
Minors: bicubic maps
e
reduce e
Tutte, Philips Res. Repts 30 (1975) 205–219.
![Page 75: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/75.jpg)
Minors: bicubic maps
e reduce e
Tutte, Philips Res. Repts 30 (1975) 205–219.
![Page 76: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/76.jpg)
Minors: bicubic maps
e reduce e
Tutte, Philips Res. Repts 30 (1975) 205–219.
![Page 77: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/77.jpg)
Minors: bicubic maps
e reduce e
Tutte, Philips Res. Repts 30 (1975) 205–219.
![Page 78: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/78.jpg)
Relationships
triangulated triangle
l
alternating dimaps
l
bicubic map (reduction: Tutte 1975)
l duality
Eulerian triangulation
(reduction, in inverse form . . . : Batagelj, 1989)
l (Cavenagh & Lisoneck, 2008)
spherical latin bitrade
![Page 79: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/79.jpg)
Relationships
triangulated triangle
l
alternating dimaps
l
bicubic map (reduction: Tutte 1975)
l duality
Eulerian triangulation (reduction, in inverse form . . . : Batagelj, 1989)
l (Cavenagh & Lisoneck, 2008)
spherical latin bitrade
![Page 80: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/80.jpg)
Relationships
triangulated triangle
l
alternating dimaps
l
bicubic map (reduction: Tutte 1975)
l duality
Eulerian triangulation (reduction, in inverse form . . . : Batagelj, 1989)
l (Cavenagh & Lisoneck, 2008)
spherical latin bitrade
![Page 81: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/81.jpg)
Ultraloops, triloops, semiloops
ultraloop
1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 82: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/82.jpg)
Ultraloops, triloops, semiloops
ultraloop 1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 83: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/83.jpg)
Ultraloops, triloops, semiloops
ultraloop 1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 84: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/84.jpg)
Ultraloops, triloops, semiloops
ultraloop 1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 85: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/85.jpg)
Ultraloops, triloops, semiloops
ultraloop 1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 86: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/86.jpg)
Ultraloops, triloops, semiloops
ultraloop 1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 87: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/87.jpg)
Ultraloops, triloops, semiloops
ultraloop 1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 88: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/88.jpg)
Ultraloops, triloops, semiloops
ultraloop 1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 89: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/89.jpg)
Ultraloops, triloops, semiloops
ultraloop 1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 90: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/90.jpg)
Ultraloops, triloops, semiloops
ultraloop 1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 91: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/91.jpg)
Ultraloops, triloops, semiloops
ultraloop 1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 92: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/92.jpg)
Ultraloops, triloops, semiloops
ultraloop 1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 93: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/93.jpg)
Ultraloops, triloops, semiloops
ultraloop 1-loop
ω-loop
ω2-loop
1-semiloop
ω-semiloop
ω2-semiloop
![Page 94: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/94.jpg)
Ultraloops, triloops, semiloops: the bicubic map
trihedron(ultraloop)
e
digon
(triloop)
e
(semiloop)
e
![Page 95: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/95.jpg)
Ultraloops, triloops, semiloops: the bicubic map
trihedron(ultraloop)
e
digon
(triloop)
e
(semiloop)
e
![Page 96: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/96.jpg)
Ultraloops, triloops, semiloops: the bicubic map
trihedron(ultraloop)
e
digon
(triloop)
e
(semiloop)
e
![Page 97: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/97.jpg)
Non-commutativity
Some bad news: sometimes,
G [µ]e[ν]f 6= G [ν]f [µ]e
![Page 98: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/98.jpg)
f
e
G
G [ω]f [1]e G [1]e[ω]f
![Page 99: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/99.jpg)
f
e
G
G [ω]f [1]e
G [1]e[ω]f
![Page 100: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/100.jpg)
f
e
G
G [ω]f [1]e G [1]e[ω]f
![Page 101: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/101.jpg)
f
e
G [ω]f [1]e 6= G [1]e[ω]f
TheoremExcept for the above situation and its trials, reductions commute.
G [µ]f [ν]e = G [ν]e[µ]f
Corollary
If µ = ν, or one of e, f is a triloop, then reductions commute.
![Page 102: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/102.jpg)
f
e
G [ω]f [1]e 6= G [1]e[ω]f
TheoremExcept for the above situation and its trials, reductions commute.
G [µ]f [ν]e = G [ν]e[µ]f
Corollary
If µ = ν, or one of e, f is a triloop, then reductions commute.
![Page 103: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/103.jpg)
Trimedial graph
G
tri(G )
TheoremAll pairs of reductions on Gcommute if and only if thetriloops of G form a vertexcover in tri(G ).
u
v
e
w1 w2
![Page 104: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/104.jpg)
Trimedial graph
G
tri(G )
TheoremAll pairs of reductions on Gcommute if and only if thetriloops of G form a vertexcover in tri(G ).
u
v
e
w1 w2
![Page 105: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/105.jpg)
Trimedial graph
G
tri(G )
TheoremAll pairs of reductions on Gcommute if and only if thetriloops of G form a vertexcover in tri(G ).
u
v
e
w1 w2
![Page 106: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/106.jpg)
Trimedial graph
G
tri(G )
TheoremAll pairs of reductions on Gcommute if and only if thetriloops of G form a vertexcover in tri(G ).
u
v
e
w1 w2
![Page 107: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/107.jpg)
Trimedial graph
G
tri(G )
TheoremAll pairs of reductions on Gcommute if and only if thetriloops of G form a vertexcover in tri(G ).
u
v
e
w1 w2
![Page 108: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/108.jpg)
Trimedial graph
G
tri(G )
TheoremAll pairs of reductions on Gcommute if and only if thetriloops of G form a vertexcover in tri(G ).
u
v
e
w1 w2
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Non-commutativity
TheoremAll sequences of reductions on G commute if and only if eachcomponent of G has the form . . .
![Page 110: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/110.jpg)
Non-commutativity
TheoremAll sequences of reductions on G commute if and only if eachcomponent of G has the form . . .
![Page 111: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/111.jpg)
Non-commutativity
ProblemCharacterise alternating dimaps such that all pairs of reductionscommute up to isomorphism:
∀µ, ν, e, f : G [µ]f [ν]e ∼= G [ν]e[µ]f
![Page 112: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/112.jpg)
Excluded minors for bounded genus
k-posy :An alternating dimap with . . .
I one vertex,
I 2k + 1 edges,
I two faces.
V − E + F = 1− (2k + 1) + 2 = 2− 2k
Genus of k-posy = k
TheoremA nonempty alternating dimap G has genus < k if and only if noneof its minors is a disjoint union of posies of total genus k.
cf. Courcelle & Dussaux (2002): ordinary maps, surface minors, bouquets.
![Page 113: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/113.jpg)
Excluded minors for bounded genus
k-posy :An alternating dimap with . . .
I one vertex,
I 2k + 1 edges,
I two faces.
V − E + F = 1− (2k + 1) + 2 = 2− 2k
Genus of k-posy = k
TheoremA nonempty alternating dimap G has genus < k if and only if noneof its minors is a disjoint union of posies of total genus k.
cf. Courcelle & Dussaux (2002): ordinary maps, surface minors, bouquets.
![Page 114: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/114.jpg)
Excluded minors for bounded genus
0-posy:
a
a
![Page 115: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/115.jpg)
Excluded minors for bounded genus
0-posy:
a
a
![Page 116: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/116.jpg)
Excluded minors for bounded genus
1-posy:
a
b
c
a
b
c
![Page 117: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/117.jpg)
Excluded minors for bounded genus
2-posy: first:
a
b
c
d
e
a
b
c
d
e
![Page 118: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/118.jpg)
Excluded minors for bounded genus
2-posy: second:
a
c
d
b
e
a
b
e
c
d
![Page 119: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/119.jpg)
Excluded minors for bounded genus
2-posy: third:
a
b
c
d
b
e
d
a
e
c
![Page 120: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/120.jpg)
Excluded minors for bounded genus
TheoremA nonempty alternating dimap G has genus < k if and only if noneof its minors is a disjoint union of posies of total genus k.
Proof.
(=⇒) Easy.
(⇐=) Show:
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Induction on |E (G )|.
Inductive basis:|E (G )| = 1 =⇒ G is an ultraloop =⇒ 0-posy minor.
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Excluded minors for bounded genus
TheoremA nonempty alternating dimap G has genus < k if and only if noneof its minors is a disjoint union of posies of total genus k.
Proof.
(=⇒) Easy.
(⇐=) Show:
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Induction on |E (G )|.
Inductive basis:|E (G )| = 1 =⇒ G is an ultraloop =⇒ 0-posy minor.
![Page 122: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/122.jpg)
Excluded minors for bounded genus
TheoremA nonempty alternating dimap G has genus < k if and only if noneof its minors is a disjoint union of posies of total genus k.
Proof.
(=⇒) Easy.
(⇐=) Show:
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Induction on |E (G )|.
Inductive basis:|E (G )| = 1 =⇒ G is an ultraloop =⇒ 0-posy minor.
![Page 123: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/123.jpg)
Excluded minors for bounded genus
TheoremA nonempty alternating dimap G has genus < k if and only if noneof its minors is a disjoint union of posies of total genus k.
Proof.
(=⇒) Easy.
(⇐=) Show:
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Induction on |E (G )|.
Inductive basis:|E (G )| = 1 =⇒ G is an ultraloop =⇒ 0-posy minor.
![Page 124: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/124.jpg)
Excluded minors for bounded genus
TheoremA nonempty alternating dimap G has genus < k if and only if noneof its minors is a disjoint union of posies of total genus k.
Proof.
(=⇒) Easy.
(⇐=) Show:
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Induction on |E (G )|.
Inductive basis:|E (G )| = 1 =⇒ G is an ultraloop =⇒ 0-posy minor.
![Page 125: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/125.jpg)
Excluded minors for bounded genus
TheoremA nonempty alternating dimap G has genus < k if and only if noneof its minors is a disjoint union of posies of total genus k.
Proof.
(=⇒) Easy.
(⇐=) Show:
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Induction on |E (G )|.
Inductive basis:|E (G )| = 1 =⇒ G is an ultraloop =⇒ 0-posy minor.
![Page 126: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/126.jpg)
Excluded minors for bounded genus
Showing . . .
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Inductive step: Suppose true for alt. dimaps of < m edges.Let G be an alternating dimap with |E (G )| = m.
G [1]e, G [ω]e, G [ω2]e each have m − 1 edges.∴ by inductive hypothesis, these each have, as a minor, a disjointunion of posies of total genus . . .γ(G [1]e), γ(G [ω]e), γ(G [ω2]e), respectively.If any of these = γ(G ): done.It remains to consider:γ(G [1]e) = γ(G [ω]e) = γ(G [ω2]e) = γ(G )− 1.
↑ ↑ ↑proper proper proper
1-semiloop ω-semiloop ω2-semiloop
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Excluded minors for bounded genus
Showing . . .
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Inductive step: Suppose true for alt. dimaps of < m edges.
Let G be an alternating dimap with |E (G )| = m.G [1]e, G [ω]e, G [ω2]e each have m − 1 edges.
∴ by inductive hypothesis, these each have, as a minor, a disjointunion of posies of total genus . . .γ(G [1]e), γ(G [ω]e), γ(G [ω2]e), respectively.If any of these = γ(G ): done.It remains to consider:γ(G [1]e) = γ(G [ω]e) = γ(G [ω2]e) = γ(G )− 1.
↑ ↑ ↑proper proper proper
1-semiloop ω-semiloop ω2-semiloop
![Page 128: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/128.jpg)
Excluded minors for bounded genus
Showing . . .
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Inductive step: Suppose true for alt. dimaps of < m edges.Let G be an alternating dimap with |E (G )| = m.
G [1]e, G [ω]e, G [ω2]e each have m − 1 edges.∴ by inductive hypothesis, these each have, as a minor, a disjointunion of posies of total genus . . .γ(G [1]e), γ(G [ω]e), γ(G [ω2]e), respectively.If any of these = γ(G ): done.It remains to consider:γ(G [1]e) = γ(G [ω]e) = γ(G [ω2]e) = γ(G )− 1.
↑ ↑ ↑proper proper proper
1-semiloop ω-semiloop ω2-semiloop
![Page 129: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/129.jpg)
Excluded minors for bounded genus
Showing . . .
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Inductive step: Suppose true for alt. dimaps of < m edges.Let G be an alternating dimap with |E (G )| = m.
G [1]e, G [ω]e, G [ω2]e each have m − 1 edges.
∴ by inductive hypothesis, these each have, as a minor, a disjointunion of posies of total genus . . .γ(G [1]e), γ(G [ω]e), γ(G [ω2]e), respectively.If any of these = γ(G ): done.It remains to consider:γ(G [1]e) = γ(G [ω]e) = γ(G [ω2]e) = γ(G )− 1.
↑ ↑ ↑proper proper proper
1-semiloop ω-semiloop ω2-semiloop
![Page 130: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/130.jpg)
Excluded minors for bounded genus
Showing . . .
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Inductive step: Suppose true for alt. dimaps of < m edges.Let G be an alternating dimap with |E (G )| = m.
G [1]e, G [ω]e, G [ω2]e each have m − 1 edges.∴ by inductive hypothesis, these each have, as a minor, a disjointunion of posies of total genus . . .γ(G [1]e), γ(G [ω]e), γ(G [ω2]e), respectively.
If any of these = γ(G ): done.It remains to consider:γ(G [1]e) = γ(G [ω]e) = γ(G [ω2]e) = γ(G )− 1.
↑ ↑ ↑proper proper proper
1-semiloop ω-semiloop ω2-semiloop
![Page 131: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/131.jpg)
Excluded minors for bounded genus
Showing . . .
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Inductive step: Suppose true for alt. dimaps of < m edges.Let G be an alternating dimap with |E (G )| = m.
G [1]e, G [ω]e, G [ω2]e each have m − 1 edges.∴ by inductive hypothesis, these each have, as a minor, a disjointunion of posies of total genus . . .γ(G [1]e), γ(G [ω]e), γ(G [ω2]e), respectively.If any of these = γ(G ): done.
It remains to consider:γ(G [1]e) = γ(G [ω]e) = γ(G [ω2]e) = γ(G )− 1.
↑ ↑ ↑proper proper proper
1-semiloop ω-semiloop ω2-semiloop
![Page 132: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/132.jpg)
Excluded minors for bounded genus
Showing . . .
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Inductive step: Suppose true for alt. dimaps of < m edges.Let G be an alternating dimap with |E (G )| = m.
G [1]e, G [ω]e, G [ω2]e each have m − 1 edges.∴ by inductive hypothesis, these each have, as a minor, a disjointunion of posies of total genus . . .γ(G [1]e), γ(G [ω]e), γ(G [ω2]e), respectively.If any of these = γ(G ): done.It remains to consider:γ(G [1]e) = γ(G [ω]e) = γ(G [ω2]e) = γ(G )− 1.
↑ ↑ ↑proper proper proper
1-semiloop ω-semiloop ω2-semiloop
![Page 133: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/133.jpg)
Excluded minors for bounded genus
Showing . . .
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Inductive step: Suppose true for alt. dimaps of < m edges.Let G be an alternating dimap with |E (G )| = m.
G [1]e, G [ω]e, G [ω2]e each have m − 1 edges.∴ by inductive hypothesis, these each have, as a minor, a disjointunion of posies of total genus . . .γ(G [1]e), γ(G [ω]e), γ(G [ω2]e), respectively.If any of these = γ(G ): done.It remains to consider:γ(G [1]e) = γ(G [ω]e) = γ(G [ω2]e) = γ(G )− 1.
↑
↑ ↑
proper
proper proper
1-semiloop
ω-semiloop ω2-semiloop
![Page 134: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/134.jpg)
Excluded minors for bounded genus
Showing . . .
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Inductive step: Suppose true for alt. dimaps of < m edges.Let G be an alternating dimap with |E (G )| = m.
G [1]e, G [ω]e, G [ω2]e each have m − 1 edges.∴ by inductive hypothesis, these each have, as a minor, a disjointunion of posies of total genus . . .γ(G [1]e), γ(G [ω]e), γ(G [ω2]e), respectively.If any of these = γ(G ): done.It remains to consider:γ(G [1]e) = γ(G [ω]e) = γ(G [ω2]e) = γ(G )− 1.
↑
↑
↑proper
proper
proper1-semiloop
ω-semiloop
ω2-semiloop
![Page 135: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/135.jpg)
Excluded minors for bounded genus
Showing . . .
γ(G ) ≥ k =⇒ ∃minor ∼= disjoint union of posies, total genus k.
Inductive step: Suppose true for alt. dimaps of < m edges.Let G be an alternating dimap with |E (G )| = m.
G [1]e, G [ω]e, G [ω2]e each have m − 1 edges.∴ by inductive hypothesis, these each have, as a minor, a disjointunion of posies of total genus . . .γ(G [1]e), γ(G [ω]e), γ(G [ω2]e), respectively.If any of these = γ(G ): done.It remains to consider:γ(G [1]e) = γ(G [ω]e) = γ(G [ω2]e) = γ(G )− 1.
↑ ↑
↑
proper proper
proper
1-semiloop ω-semiloop
ω2-semiloop
![Page 136: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/136.jpg)
Excluded minors for bounded genus
F1
F1
F1
F2
F2
F2
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Excluded minors for bounded genus
F1
F1
F1
F2
F2
F2
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Excluded minors for bounded genus
F1
F1
F1
F2
F2
F2
![Page 139: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/139.jpg)
Excluded minors for bounded genus
F1
F1
F1
F2
F2
F2
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Excluded minors for bounded genus
F1
F1
F1
F2
F2
F2
![Page 141: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/141.jpg)
Excluded minors for bounded genus
F1
F1
F1
F2
F2
F2
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Excluded minors for bounded genus
F1
F1
F1
F2
F2
F2
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Excluded minors for bounded genus
F1
F1
F1
F2
F2
F2
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Tutte polynomial of a graph (or matroid)
T (G ; x , y) =∑X⊆E
(x − 1)ρ(E)−ρ(X )(y − 1)ρ∗(E)−ρ∗(E\X )
where
ρ(Y ) = rank of Y
= (#vertices that meet Y )− (# components of Y ),
ρ∗(Y ) = rank of Y in the dual, G ∗
= |X |+ ρ(E \ X )− ρ(E ).
By appropriate substitutions, it yields:
numbers of colourings, acyclic orientations, spanning trees,spanning subgraphs, forests, . . .chromatic polynomial, flow polynomial, reliability polynomial,Ising and Potts model partition functions, weight enumeratorof a linear code, Jones polynomial of an alternating link, ...
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Tutte polynomial of a graph (or matroid)
T (G ; x , y) =∑X⊆E
(x − 1)ρ(E)−ρ(X )(y − 1)ρ∗(E)−ρ∗(E\X )
where
ρ(Y ) = rank of Y
= (#vertices that meet Y )− (# components of Y ),
ρ∗(Y ) = rank of Y in the dual, G ∗
= |X |+ ρ(E \ X )− ρ(E ).
By appropriate substitutions, it yields:
numbers of colourings, acyclic orientations, spanning trees,spanning subgraphs, forests, . . .
chromatic polynomial, flow polynomial, reliability polynomial,Ising and Potts model partition functions, weight enumeratorof a linear code, Jones polynomial of an alternating link, ...
![Page 146: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/146.jpg)
Tutte polynomial of a graph (or matroid)
T (G ; x , y) =∑X⊆E
(x − 1)ρ(E)−ρ(X )(y − 1)ρ∗(E)−ρ∗(E\X )
where
ρ(Y ) = rank of Y
= (#vertices that meet Y )− (# components of Y ),
ρ∗(Y ) = rank of Y in the dual, G ∗
= |X |+ ρ(E \ X )− ρ(E ).
By appropriate substitutions, it yields:
numbers of colourings, acyclic orientations, spanning trees,spanning subgraphs, forests, . . .chromatic polynomial, flow polynomial, reliability polynomial,Ising and Potts model partition functions, weight enumeratorof a linear code, Jones polynomial of an alternating link, ...
![Page 147: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/147.jpg)
Tutte polynomial of a graph (or matroid)Deletion-contraction relation:T (G ; x , y) =
1, if G is empty,x T (G \ e; x , y), if e is a coloop (i.e., bridge),y T (G/e; x , y), if e is a loop,T (G \ e; x , y) + T (G/e; x , y), if e is neither a coloop nor a loop.
Recipe Theorem (in various forms: Tutte, 1948; Brylawski, 1972;Oxley & Welsh, 1979):If F is an isomorphism invariant and satisfies . . .F (G ) =
x F (G \ e), if e is a coloop (i.e., bridge),y F (G/e), if e is a loop,a F (G \ e) + b F (G/e), if e is neither a coloop nor a loop.
. . . then it can be obtained from the Tutte polynomial usingappropriate substitutions and factors.
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Tutte polynomial of a graph (or matroid)Deletion-contraction relation:T (G ; x , y) =
1, if G is empty,x T (G \ e; x , y), if e is a coloop (i.e., bridge),y T (G/e; x , y), if e is a loop,T (G \ e; x , y) + T (G/e; x , y), if e is neither a coloop nor a loop.
Recipe Theorem (in various forms: Tutte, 1948; Brylawski, 1972;Oxley & Welsh, 1979):If F is an isomorphism invariant and satisfies . . .F (G ) =
x F (G \ e), if e is a coloop (i.e., bridge),y F (G/e), if e is a loop,a F (G \ e) + b F (G/e), if e is neither a coloop nor a loop.
. . . then it can be obtained from the Tutte polynomial usingappropriate substitutions and factors.
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Tutte invariant for alternating dimaps
– an isomorphism invariant F such that:F (G ) =
1, if G is empty,w F (G − e), if e is an ultraloop,x F (G [1]e), if e is a proper 1-loop,y F (G [ω]e), if e is a proper ω-loop,z F (G [ω2]e), if e is a proper ω2-loop,a F (G [1]e) + b F (G [ω]e) + c F (G [ω2]e), if e is not a triloop.
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Tutte invariant for alternating dimaps
TheoremThe only Tutte invariants of alternating dimaps are:
(a) F (G ) = 0 for nonempty G ,
(b) F (G ) = 3|E(G)|a|V (G)|bc-faces(G)ca-faces(G),
(c) F (G ) = a|V (G)|bc-faces(G)(−c)a-faces(G),
(d) F (G ) = a|V (G)|(−b)c-faces(G)ca-faces(G),
(e) F (G ) = (−a)|V (G)|bc-faces(G)ca-faces(G).
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Extended Tutte invariant for alternating dimaps
– an isomorphism invariant F such that:
F (G ) =
1, if G is empty,w F (G − e), if e is an ultraloop,x F (G [1]e), if e is a proper 1-loop,y F (G [ω]e), if e is a proper ω-loop,z F (G [ω2]e), if e is a proper ω2-loop,a F (G [1]e) + b F (G [ω]e) + c F (G [ω2]e), if e is a proper 1-semiloop,d F (G [1]e) + e F (G [ω]e) + f F (G [ω2]e), if e is a proper ω-semiloop,g F (G [1]e) + h F (G [ω]e) + i F (G [ω2]e), if e is a proper ω2-semiloop,j F (G [1]e) + k F (G [ω]e) + l F (G [ω2]e), if e is not a triloop.
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Extended Tutte invariant for alternating dimapsFor any alternating dimap G , define Tc(G ; x , y) and Ta(G ; x , y) as
follows.
Tc(G ; x , y)
=
1, if G is empty,Tc(G [∗]e; x , y), if e is an ω2-loop;x Tc(G [ω2]e; x , y), if e is an ω-semiloop;y Tc(G [1]e; x , y), if e is a proper 1-semiloop or an ω-loop;Tc(G [1]e; x , y) + Tc(G [ω2]e; x , y), if e is not a semiloop.
Ta(G ; x , y)
=
1, if G is empty,Ta(G [∗]e; x , y), if e is an ω-loop;x Ta(G [ω]e; x , y), if e is an ω2-semiloop;y Ta(G [1]e; x , y), if e is a proper 1-semiloop or an ω2-loop;Ta(G [1]e; x , y) + Ta(G [ω]e; x , y), if e is not a semiloop.
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Extended Tutte invariant for alternating dimaps
TheoremFor any plane graph G,
T (G ; x , y) = Tc(altc(G ); x , y)
= Ta(alta(G ); x , y).
u
v
u
v
u
v
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Extended Tutte invariant for alternating dimaps
TheoremFor any plane graph G,
T (G ; x , y) = Tc(altc(G ); x , y)
= Ta(alta(G ); x , y).
u
v
u
v
u
v
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Extended Tutte invariant for alternating dimaps
TheoremFor any plane graph G,
T (G ; x , y) = Tc(altc(G ); x , y)
= Ta(alta(G ); x , y).
u
v
u
v
u
v
![Page 156: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/156.jpg)
Extended Tutte invariant for alternating dimaps
TheoremFor any plane graph G,
T (G ; x , y) = Tc(altc(G ); x , y) = Ta(alta(G ); x , y).
u
v
u
v
u
v
![Page 157: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/157.jpg)
Extended Tutte invariant for alternating dimaps
TheoremFor any plane graph G,
T (G ; x , y) = Tc(altc(G ); x , y) = Ta(alta(G ); x , y).
u
v
u
v
u
v
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Extended Tutte invariant for alternating dimaps
Ti (G ; x) =
1, if G is empty,Ti (G [∗]e; x), if e is a 1-loop (including an ultraloop);x Ti (G [ω2]e; x), if e is a proper ω-semiloop or an ω2-loop;x Ti (G [ω]e; x), if e is a proper ω2-semiloop or an ω-loop;Ti (G [ω]e; x) + Ti (G [ω2]e; x), if e is not a semiloop.
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Extended Tutte invariant for alternating dimaps
TheoremFor any plane graph G,
T (G ; x , x) = Ti (alti(G ); x).
u
v
![Page 160: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/160.jpg)
Extended Tutte invariant for alternating dimaps
TheoremFor any plane graph G,
T (G ; x , x) = Ti (alti(G ); x).
u
v
![Page 161: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/161.jpg)
Extended Tutte invariant for alternating dimaps
TheoremFor any plane graph G,
T (G ; x , x) = Ti (alti(G ); x).
u
v
![Page 162: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/162.jpg)
Extended Tutte invariant for alternating dimaps
TheoremFor any plane graph G,
T (G ; x , x) = Ti (alti(G ); x).
u
v
![Page 163: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/163.jpg)
Extended Tutte invariant for alternating dimaps
TheoremFor any plane graph G,
T (G ; x , x) = Ti (alti(G ); x).
u
v
![Page 164: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/164.jpg)
Extended Tutte invariant for alternating dimaps
TheoremFor any plane graph G,
T (G ; x , x) = Ti (alti(G ); x).
u
v
![Page 165: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/165.jpg)
References
I R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte,The dissection of rectangles into squares, Duke Math. J. 7(1940) 312–340.
I W. T. Tutte, The dissection of equilateral triangles intoequilateral triangles, Proc. Cambridge Philos. Soc. 44 (1948)463–482.
I W. T. Tutte, Duality and trinity, in: Infinite and Finite Sets(Colloq., Keszthely, 1973), Vol. III, Colloq. Math. Soc. JanosBolyai, Vol. 10, North-Holland, Amsterdam, 1975, pp.1459–1472.
I R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte,Leaky electricity and triangulated triangles, Philips Res. Repts.30 (1975) 205–219.
I W. T. Tutte, Bicubic planar maps, Symposium a la Memoirede Francois Jaeger (Grenoble, 1998), Ann. Inst. Fourier(Grenoble) 49 (1999) 1095–1102.
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References
For more information:
I GF, Minors for alternating dimaps, preprint, 2013,http://arxiv.org/abs/1311.2783.
I GF, Transforms and minors for binary functions,Ann. Combin. 17 (2013) 477–493.
For less information:
I GF, short public talk (10 mins) on ‘William Tutte’,The Laborastory, 2013,https:
//soundcloud.com/thelaborastory/william-thomas-tutte
![Page 167: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/167.jpg)
References
For more information:
I GF, Minors for alternating dimaps, preprint, 2013,http://arxiv.org/abs/1311.2783.
I GF, Transforms and minors for binary functions,Ann. Combin. 17 (2013) 477–493.
For less information:
I GF, short public talk (10 mins) on ‘William Tutte’,The Laborastory, 2013,https:
//soundcloud.com/thelaborastory/william-thomas-tutte
![Page 168: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/168.jpg)
References
For more information:
I GF, Minors for alternating dimaps, preprint, 2013,http://arxiv.org/abs/1311.2783.
I GF, Transforms and minors for binary functions,Ann. Combin. 17 (2013) 477–493.
For less information:
I GF, short public talk (10 mins) on ‘William Tutte’,The Laborastory, 2013,https:
//soundcloud.com/thelaborastory/william-thomas-tutte
![Page 169: Minors and Tutte invariants for alternating dimapsusers.monash.edu/~gfarr/research/slides/Farr-alt-dimap-talk-2014.pdf · The order doesn’t matter. Deletion and contraction commute:](https://reader034.fdocuments.net/reader034/viewer/2022050612/5fb316215f64f04b7c7479d9/html5/thumbnails/169.jpg)
References
For more information:
I GF, Minors for alternating dimaps, preprint, 2013,http://arxiv.org/abs/1311.2783.
I GF, Transforms and minors for binary functions,Ann. Combin. 17 (2013) 477–493.
For less information:
I GF, short public talk (10 mins) on ‘William Tutte’,The Laborastory, 2013,https:
//soundcloud.com/thelaborastory/william-thomas-tutte