Minors and Tutte invariants for alternating dimaps

Transcription

1 Minors and Tutte invariants for alternating dimaps Graham Farr Clayton School of IT Monash University Work done partly at: Isaac Newton Institute for Mathematical Sciences (Combinatorics and Statistical Mechanics Programme), Cambridge, 2008; University of Melbourne (sabbatical), 2011; and Queen Mary, University of London, March 2014

2 Contraction and Deletion G u e v G/e G \ e u = v u v

3 Minors H is a minor of G if it can be obtained from G by some sequence of 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

4 Minors H is a minor of G if it can be obtained from G by some sequence of deletions and/or contractions. The order doesn t matter. Deletion and contraction commute: Importance of minors: G/e/f = G/f /e G \ e \ f = G \ f \ e G/e \ f = G \ f /e excluded minor characterisations planar graphs (Kuratowski, 1930; Wagner, 1937) graphs, among matroids (Tutte, PhD thesis, 1948) Robertson-Seymour Theorem ( ) counting Tutte-Whitney polynomial family

5 Duality and minors Classical duality for embedded graphs: G G vertices faces

6 Duality and minors Classical duality for embedded graphs: G G vertices faces contraction deletion (G/e) = G \ e (G \ e) = G /e

7 Duality and minors G G \ e G/e

8 Duality and minors G G G \ e G/e

9 Duality and minors G G G \ e G \ e G/e G /e

10 Duality and minors G G G \ e G \ e G/e G /e

11 Loops and coloops loop coloop = bridge = isthmus

12 Loops and coloops loop coloop = bridge = isthmus duality

13 History H. E. Dudeney, Puzzling Times at Solvamhall Castle: Lady Isabel s Casket, London Magazine 7 (42) (Jan 1902) 584

14 History London Magazine 8 (43) (Feb 1902) 56

15 History First published by Heinemann, London, Above is from 4th edn, Nelson, 1932.

16 History Duke Math. J. 7 (1940)

17 History from a design for a proposed memorial to Tutte in Newmarket, UK.

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31 History Proc. Cambridge Philos. Soc. 44 (1948)

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41 triad of alternating dimaps

42 triad of alternating dimaps

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45 bicubic map

46 bicubic map

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51 Alternating dimaps Alternating dimap (Tutte, 1948): directed graph without isolated vertices, 2-cell embedded in a disjoint union of orientable 2-manifolds, each vertex has even degree, v: edges incident with v are directed alternately into, and out of, v (as you go around v).

52 Alternating dimaps Alternating dimap (Tutte, 1948): directed graph without isolated vertices, 2-cell embedded in a disjoint union of orientable 2-manifolds, each vertex has even degree, v: edges incident with v are directed alternately into, and out of, v (as you go around v). So vertices look like this:

53 Alternating dimaps Alternating dimap (Tutte, 1948): directed graph without isolated vertices, 2-cell embedded in a disjoint union of orientable 2-manifolds, each vertex has even degree, 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))

54 Alternating dimaps Three special partitions of E(G): clockwise faces anticlockwise faces in-stars (An in-star is the set of all edges going into some vertex.)

55 Alternating dimaps Three special partitions of E(G): clockwise faces anticlockwise faces in-stars (An in-star is the set of all edges going into some vertex.) Each defines a permutation of E(G).

56 Alternating dimaps Three special partitions of E(G): clockwise faces σ c anticlockwise faces σ a in-stars σ i (An in-star is the set of all edges going into some vertex.) Each defines a permutation of E(G).

57 Alternating dimaps Three special partitions of E(G): clockwise faces σ c anticlockwise faces σ a in-stars σ i (An in-star is the set of all edges going into some vertex.) Each defines a permutation of E(G). These permutations satisfy σ i σ c σ a = 1

58 Triality (Trinity) Construction of trial map: clockwise faces vertices anticlockwise faces clockwise faces

59 Triality (Trinity) Construction of trial map: clockwise faces vertices anticlockwise faces clockwise faces (σ i, σ c, σ a ) (σ c, σ a, σ i )

60 Triality (Trinity) Construction of trial map: clockwise faces vertices anticlockwise faces clockwise faces (σ i, σ c, σ a ) (σ c, σ a, σ i ) e f u

61 Triality (Trinity) Construction of trial map: clockwise faces vertices anticlockwise faces clockwise faces (σ i, σ c, σ a ) (σ c, σ a, σ i ) v C1 e e ω f v C2 u

62 Minor operations u G e v w 1 w 2

63 Minor operations G[1]e u = v w 1 w 2

64 Minor operations u G e v w 1 w 2

65 Minor operations u G[ω]e v w 1 w 2

66 Minor operations u G e v w 1 w 2

67 Minor operations u G[ω 2 ]e w 1 w 2 v

68 Minor operations u G e v w 1 w 2

69 Minor operations u G e v e ω w 1 w 2

70 Minor operations G[1]e u = v w 1 w 2

71 Minor operations (G[1]e) ω = G ω [ω 2 ]e ω u = v w 1 w 2

72 Minor operations G ω [1]e ω = (G[ω]e) ω, G ω [ω]e ω = (G[ω 2 ]e) ω, G ω [ω 2 ]e ω = (G[1]e) ω, G ω2 [1]e ω2 = (G[ω 2 ]e) ω2, G ω2 [ω]e ω2 = (G[1]e) ω2, G ω2 [ω 2 ]e ω2 = (G[ω]e) ω2. Theorem If e E(G) and µ, ν {1, ω, ω 2 } then G µ [ν]e ω = (G[µν]e) µ. Same pattern as established for other generalised minor operations (GF, 2008/ ).

73 Minor operations G G ω G ω2 G[ω 2 ]e G ω [ω 2 ]e G ω2 [ω 2 ]e G[ω]e G ω [ω]e G ω2 [ω]e G[1]e G ω [1]e G ω2 [1]e

74 Minors: bicubic maps e

75 Minors: bicubic maps e reduce e

76 Minors: bicubic maps e reduce e

77 Minors: bicubic maps e reduce e Tutte, Philips Res. Repts 30 (1975)

78 Relationships triangulated triangle alternating dimaps bicubic map (reduction: Tutte 1975) duality Eulerian triangulation

79 Relationships triangulated triangle alternating dimaps bicubic map (reduction: Tutte 1975) duality Eulerian triangulation (reduction, in inverse form... : Batagelj, 1989)

80 Relationships triangulated triangle alternating dimaps bicubic map (reduction: Tutte 1975) duality Eulerian triangulation (reduction, in inverse form... : Batagelj, 1989) (Cavenagh & Lisoněck, 2008) spherical latin bitrade

81 Ultraloops, triloops, semiloops ultraloop

82 Ultraloops, triloops, semiloops ultraloop 1-loop

83 Ultraloops, triloops, semiloops ω-loop ultraloop 1-loop

84 Ultraloops, triloops, semiloops ω-loop ultraloop 1-loop ω 2 -loop

85 Ultraloops, triloops, semiloops ω-loop ultraloop 1-loop ω 2 -loop

86 Ultraloops, triloops, semiloops ω-loop ultraloop 1-loop ω 2 -loop

87 Ultraloops, triloops, semiloops ω-loop ultraloop 1-loop ω 2 -loop

88 Ultraloops, triloops, semiloops ω-loop 1-semiloop ultraloop 1-loop ω 2 -loop

89 Ultraloops, triloops, semiloops ω-loop 1-semiloop ultraloop 1-loop ω 2 -loop ω-semiloop

90 Ultraloops, triloops, semiloops ω 2 -semiloop ω-loop 1-semiloop ultraloop 1-loop ω 2 -loop ω-semiloop

91 Ultraloops, triloops, semiloops ω 2 -semiloop ω-loop 1-semiloop ultraloop 1-loop ω 2 -loop ω-semiloop

92 Ultraloops, triloops, semiloops ω 2 -semiloop ω-loop 1-semiloop ultraloop 1-loop ω 2 -loop ω-semiloop

93 Ultraloops, triloops, semiloops ω 2 -semiloop ω-loop 1-semiloop ultraloop 1-loop ω 2 -loop ω-semiloop

94 Ultraloops, triloops, semiloops: the bicubic map trihedron (ultraloop) e

95 Ultraloops, triloops, semiloops: the bicubic map trihedron (ultraloop) digon (triloop) e e

96 Ultraloops, triloops, semiloops: the bicubic map trihedron (ultraloop) digon (triloop) (semiloop) e e e

97 Non-commutativity Some bad news: sometimes, G[µ]e[ν]f G[ν]f [µ]e

98 f e G

99 f e G G[ω]f [1]e

100 f e G G[ω]f [1]e G[1]e[ω]f

101 f e G[ω]f [1]e G[1]e[ω]f

102 f e G[ω]f [1]e G[1]e[ω]f Theorem Except 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.

103 Trimedial graph G u e v w 1 w 2

104 Trimedial graph G u tri(g) e v w 1 w 2

105 Trimedial graph G u tri(g) e v w 1 w 2

106 Trimedial graph G u tri(g) e v w 1 w 2

107 Trimedial graph G u tri(g) e v w 1 w 2

108 Trimedial graph G u tri(g) e Theorem All pairs of reductions on G commute if and only if the triloops of G form a vertex cover in tri(g). v w 1 w 2

109 Non-commutativity Theorem All sequences of reductions on G commute if and only if each component of G has the form...

110 Non-commutativity Theorem All sequences of reductions on G commute if and only if each component of G has the form...

111 Non-commutativity Problem Characterise alternating dimaps such that all pairs of reductions commute up to isomorphism: µ, ν, e, f : G[µ]f [ν]e = G[ν]e[µ]f

112 Excluded minors for bounded genus k-posy: An alternating dimap with... one vertex, 2k + 1 edges, two faces. Genus of k-posy = k V E + F = 1 (2k + 1) + 2 = 2 2k Theorem A nonempty alternating dimap G has genus < k if and only if none of its minors is a disjoint union of posies of total genus k.

113 Excluded minors for bounded genus k-posy: An alternating dimap with... one vertex, 2k + 1 edges, two faces. Genus of k-posy = k V E + F = 1 (2k + 1) + 2 = 2 2k Theorem A nonempty alternating dimap G has genus < k if and only if none of its minors is a disjoint union of posies of total genus k. cf. Courcelle & Dussaux (2002): ordinary maps, surface minors, bouquets.

114 Excluded minors for bounded genus 0-posy:

115 Excluded minors for bounded genus a 0-posy: a

116 Excluded minors for bounded genus a c b 1-posy: b c a

117 Excluded minors for bounded genus e a b d c 2-posy: first: c d b a e

118 Excluded minors for bounded genus d a c c d 2-posy: second: e b b a e

119 Excluded minors for bounded genus c a b e c 2-posy: third: a d d e b

120 Excluded minors for bounded genus Theorem A nonempty alternating dimap G has genus < k if and only if none of its minors is a disjoint union of posies of total genus k.

121 Excluded minors for bounded genus Theorem A nonempty alternating dimap G has genus < k if and only if none of its minors is a disjoint union of posies of total genus k. Proof.

122 Excluded minors for bounded genus Theorem A nonempty alternating dimap G has genus < k if and only if none of its minors is a disjoint union of posies of total genus k. Proof. (= ) Easy.

123 Excluded minors for bounded genus Theorem A nonempty alternating dimap G has genus < k if and only if none of 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.

124 Excluded minors for bounded genus Theorem A nonempty alternating dimap G has genus < k if and only if none of 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).

125 Excluded minors for bounded genus Theorem A nonempty alternating dimap G has genus < k if and only if none of 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.

126 Excluded minors for bounded genus Showing... γ(g) k = minor = disjoint union of posies, total genus k.

127 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.

128 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.

129 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[ω 2 ]e each have m 1 edges.

130 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[ω 2 ]e each have m 1 edges. by inductive hypothesis, these each have, as a minor, a disjoint union of posies of total genus... γ(g[1]e), γ(g[ω]e), γ(g[ω 2 ]e), respectively.

131 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[ω 2 ]e each have m 1 edges. by inductive hypothesis, these each have, as a minor, a disjoint union of posies of total genus... γ(g[1]e), γ(g[ω]e), γ(g[ω 2 ]e), respectively. If any of these = γ(g): done.

132 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[ω 2 ]e each have m 1 edges. by inductive hypothesis, these each have, as a minor, a disjoint union of posies of total genus... γ(g[1]e), γ(g[ω]e), γ(g[ω 2 ]e), respectively. If any of these = γ(g): done. It remains to consider: γ(g[1]e) = γ(g[ω]e) = γ(g[ω 2 ]e) = γ(g) 1.

133 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[ω 2 ]e each have m 1 edges. by inductive hypothesis, these each have, as a minor, a disjoint union of posies of total genus... γ(g[1]e), γ(g[ω]e), γ(g[ω 2 ]e), respectively. If any of these = γ(g): done. It remains to consider: γ(g[1]e) = γ(g[ω]e) = γ(g[ω 2 ]e) = γ(g) 1. proper 1-semiloop

134 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[ω 2 ]e each have m 1 edges. by inductive hypothesis, these each have, as a minor, a disjoint union of posies of total genus... γ(g[1]e), γ(g[ω]e), γ(g[ω 2 ]e), respectively. If any of these = γ(g): done. It remains to consider: γ(g[1]e) = γ(g[ω]e) = γ(g[ω 2 ]e) = γ(g) 1. proper ω-semiloop

135 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[ω 2 ]e each have m 1 edges. by inductive hypothesis, these each have, as a minor, a disjoint union of posies of total genus... γ(g[1]e), γ(g[ω]e), γ(g[ω 2 ]e), respectively. If any of these = γ(g): done. It remains to consider: γ(g[1]e) = γ(g[ω]e) = γ(g[ω 2 ]e) = γ(g) 1. proper ω 2 -semiloop

136 Excluded minors for bounded genus

137 Excluded minors for bounded genus F 1

138 Excluded minors for bounded genus F 1 F 1

139 Excluded minors for bounded genus F 1 F 1 F 1

140 Excluded minors for bounded genus F 1 F 1 F 2 F 1

141 Excluded minors for bounded genus F 1 F 1 F 2 F 2 F 1

142 Excluded minors for bounded genus F 2 F 1 F 1 F 2 F 2 F 1

143 Excluded minors for bounded genus F 2 F 1 F 1 F 2 F 2 F 1

144 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).

145 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,...

146 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 enumerator of a linear code, Jones polynomial of an alternating link,...

147 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.

148 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 using appropriate substitutions and factors.

149 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[ω 2 ]e), if e is a proper ω 2 -loop, a F (G[1]e) + b F (G[ω]e) + c F (G[ω 2 ]e), if e is not a triloop.

150 Tutte invariant for alternating dimaps Theorem The only Tutte invariants of alternating dimaps are: (a) F (G) = 0 for nonempty G, (b) F (G) = 3 E(G) a V (G) b c-faces(g) c a-faces(g), (c) F (G) = a V (G) b c-faces(g) ( c) a-faces(g), (d) F (G) = a V (G) ( b) c-faces(g) c a-faces(g), (e) F (G) = ( a) V (G) b c-faces(g) c a-faces(g).

151 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[ω 2 ]e), if e is a proper ω 2 -loop, a F (G[1]e) + b F (G[ω]e) + c F (G[ω 2 ]e), if e is a proper 1-semiloop, d F (G[1]e) + e F (G[ω]e) + f F (G[ω 2 ]e), if e is a proper ω-semiloop, g F (G[1]e) + h F (G[ω]e) + i F (G[ω 2 ]e), if e is a proper ω 2 -semiloop, j F (G[1]e) + k F (G[ω]e) + l F (G[ω 2 ]e), if e is not a triloop.

152 Extended Tutte invariant for alternating dimaps For any alternating dimap G, define T c (G; x, y) and T a (G; x, y) as follows. T c (G; x, y) 1, if G is empty, T c (G[ ]e; x, y), if e is an ω 2 -loop; = x T c (G[ω 2 ]e; x, y), if e is an ω-semiloop; y T c (G[1]e; x, y), if e is a proper 1-semiloop or an ω-loop; T c (G[1]e; x, y) + T c (G[ω 2 ]e; x, y), if e is not a semiloop. T a (G; x, y) 1, if G is empty, T a (G[ ]e; x, y), if e is an ω-loop; = x T a (G[ω]e; x, y), if e is an ω 2 -semiloop; y T a (G[1]e; x, y), if e is a proper 1-semiloop or an ω 2 -loop; T a (G[1]e; x, y) + T a (G[ω]e; x, y), if e is not a semiloop.

153 Extended Tutte invariant for alternating dimaps Theorem For any plane graph G, T (G; x, y) = T c (alt c (G); x, y)

154 Extended Tutte invariant for alternating dimaps Theorem For any plane graph G, T (G; x, y) = T c (alt c (G); x, y) u v

155 Extended Tutte invariant for alternating dimaps Theorem For any plane graph G, T (G; x, y) = T c (alt c (G); x, y) u u v v

156 Extended Tutte invariant for alternating dimaps Theorem For any plane graph G, T (G; x, y) = T c (alt c (G); x, y) = T a (alt a (G); x, y). u u v v

157 Extended Tutte invariant for alternating dimaps Theorem For any plane graph G, T (G; x, y) = T c (alt c (G); x, y) = T a (alt a (G); x, y). u u u v v v

158 Extended Tutte invariant for alternating dimaps T i (G; x) = 1, if G is empty, T i (G[ ]e; x), if e is a 1-loop (including an ultraloop); x T i (G[ω 2 ]e; x), if e is a proper ω-semiloop or an ω 2 -loop; x T i (G[ω]e; x), if e is a proper ω 2 -semiloop or an ω-loop; T i (G[ω]e; x) + T i (G[ω 2 ]e; x), if e is not a semiloop.

159 Extended Tutte invariant for alternating dimaps Theorem For any plane graph G, T (G; x, x) = T i (alt i (G); x).

160 Extended Tutte invariant for alternating dimaps Theorem For any plane graph G, T (G; x, x) = T i (alt i (G); x). u v

161 Extended Tutte invariant for alternating dimaps Theorem For any plane graph G, T (G; x, x) = T i (alt i (G); x). u v

162 Extended Tutte invariant for alternating dimaps Theorem For any plane graph G, T (G; x, x) = T i (alt i (G); x). u v

163 Extended Tutte invariant for alternating dimaps Theorem For any plane graph G, T (G; x, x) = T i (alt i (G); x). u v

164 Extended Tutte invariant for alternating dimaps Theorem For any plane graph G, T (G; x, x) = T i (alt i (G); x). u v

165 References 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) W. T. Tutte, The dissection of equilateral triangles into equilateral triangles, Proc. Cambridge Philos. Soc. 44 (1948) W. T. Tutte, Duality and trinity, in: Infinite and Finite Sets (Colloq., Keszthely, 1973), Vol. III, Colloq. Math. Soc. Janos Bolyai, Vol. 10, North-Holland, Amsterdam, 1975, pp R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, Leaky electricity and triangulated triangles, Philips Res. Repts. 30 (1975) W. T. Tutte, Bicubic planar maps, Symposium à la Mémoire de François Jaeger (Grenoble, 1998), Ann. Inst. Fourier (Grenoble) 49 (1999)

166 References For more information:

167 References For more information: GF, Minors for alternating dimaps, preprint, 2013, GF, Transforms and minors for binary functions, Ann. Combin. 17 (2013)

168 References For more information: GF, Minors for alternating dimaps, preprint, 2013, GF, Transforms and minors for binary functions, Ann. Combin. 17 (2013) For less information:

169 References For more information: GF, Minors for alternating dimaps, preprint, 2013, GF, Transforms and minors for binary functions, Ann. Combin. 17 (2013) For less information: GF, short public talk (10 mins) on William Tutte, The Laborastory, 2013, https: //soundcloud.com/thelaborastory/william-thomas-tutte