[PPT] - Finding Tutte Paths in Linear Time Philipp Kindermann Universit PowerPoint Presentation

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Finding Tutte Paths in Linear Time

Philipp Kindermann Universit¨ at W¨ urzburg joint work with Therese Biedl University of Waterloo

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Tutte Paths

Planar graph G

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Tutte Paths

X Planar graph G

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Tutte Paths

X Planar graph G Y

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Tutte Paths

X α Planar graph G Y

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Tutte Paths

X α Tutte path: Path from X to Y via α Planar graph G Y

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Tutte Paths

X α Tutte path: Path from X to Y via α Planar graph G P Y

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Tutte Paths

X α Tutte path: Path from X to Y via α Planar graph G P Y

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Tutte Paths

X α Tutte path: Path from X to Y via α Planar graph G P Y

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Tutte Paths

X α Tutte path: Path from X to Y via α Planar graph G P Y Every comp. attached to ≤3 vtcs of P

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Tutte Paths

X α Tutte path: Path from X to Y via α Planar graph G P Y Every comp. attached to ≤3 vtcs of P

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Tutte Paths

X α Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Planar graph G P Y Every comp. attached to ≤3 vtcs of P

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Tutte Paths

X α Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Planar graph G P Y Every comp. attached to ≤3 vtcs of P

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What is known?

X Y α

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Gao, Richter & Yu ’95, ’06] G 3-conn., X, Y, α on outer face ⇒ TSDR-path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

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Tutte Paths

X Y α Planar graph G P Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P

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Tutte Paths

X Y α Planar graph G P TSDR-path: Tutte path + System of Distinct Representatives: Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P

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Tutte Paths

X Y α Planar graph G P TSDR-path: Tutte path + System of Distinct Representatives: Injective assignment of comp. to attachment pts Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P

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Tutte Paths

X Y α Planar graph G P TSDR-path: Tutte path + System of Distinct Representatives: Injective assignment of comp. to attachment pts Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P

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Tutte Paths

X Y α Planar graph G P TSDR-path: Tutte path + System of Distinct Representatives: Injective assignment of comp. to attachment pts Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Gao, Richter & Yu ’95, ’06] G 3-conn., X, Y, α on outer face ⇒ TSDR-path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Chiba & Nishizeki ’89] G 4-conn. ⇒ Tutte path in O(n) time [Gao, Richter & Yu ’95, ’06] G 3-conn., X, Y, α on outer face ⇒ TSDR-path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Chiba & Nishizeki ’89] G 4-conn. ⇒ Tutte path in O(n) time (= Hamil. path) [Gao, Richter & Yu ’95, ’06] G 3-conn., X, Y, α on outer face ⇒ TSDR-path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Chiba & Nishizeki ’89] G 4-conn. ⇒ Tutte path in O(n) time (= Hamil. path) [Schmid & Schmidt ’15] . . . in O(n2) time [Gao, Richter & Yu ’95, ’06] G 3-conn., X, Y, α on outer face ⇒ TSDR-path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Chiba & Nishizeki ’89] G 4-conn. ⇒ Tutte path in O(n) time [Schmid & Schmidt ’18] . . . in O(n2) time (= Hamil. path) [Schmid & Schmidt ’15] . . . in O(n2) time [Gao, Richter & Yu ’95, ’06] G 3-conn., X, Y, α on outer face ⇒ TSDR-path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Chiba & Nishizeki ’89] G 4-conn. ⇒ Tutte path in O(n) time [Schmid & Schmidt ’18] . . . in O(n2) time (= Hamil. path) . . . in O(n) time [Schmid & Schmidt ’15] . . . in O(n2) time [Gao, Richter & Yu ’95, ’06] G 3-conn., X, Y, α on outer face ⇒ TSDR-path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Chiba & Nishizeki ’89] G 4-conn. ⇒ Tutte path in O(n) time [Schmid & Schmidt ’18] . . . in O(n2) time (= Hamil. path) . . . in O(n) time [Schmid & Schmidt ’15] . . . in O(n2) time [Gao, Richter & Yu ’95, ’06] G 3-conn., X, Y, α on outer face ⇒ TSDR-path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Chiba & Nishizeki ’89] G 4-conn. ⇒ Tutte path in O(n) time [Schmid & Schmidt ’18] . . . in O(n2) time (= Hamil. path) . . . in O(n) time [Schmid & Schmidt ’15] . . . in O(n2) time [Gao, Richter & Yu ’95, ’06] G 3-conn., X, Y, α on outer face ⇒ TSDR-path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

Tint

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Tutte paths

X Y Planar graph G P α Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P TSDR-path: Tutte path + System of Distinct Representatives: Injective assignment of comp. to attachment pts

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Tutte paths

Tint-path: X Y Planar graph G P α Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P TSDR-path: Tutte path + System of Distinct Representatives: Injective assignment of comp. to attachment pts

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Tutte paths

Tint-path: –TSDR-path – visits all ext. vtcs – all comp. assigned to int. vtcs X Y Planar graph G P α Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P TSDR-path: Tutte path + System of Distinct Representatives: Injective assignment of comp. to attachment pts

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Tutte paths

Tint-path: –TSDR-path – visits all ext. vtcs – all comp. assigned to int. vtcs X Y Planar graph G P α Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P TSDR-path: Tutte path + System of Distinct Representatives: Injective assignment of comp. to attachment pts

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Tutte paths

Tint-path: –TSDR-path – visits all ext. vtcs – all comp. assigned to int. vtcs X Y Planar graph G P α Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P TSDR-path: Tutte path + System of Distinct Representatives: Injective assignment of comp. to attachment pts

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Tutte paths

Tint-path: –TSDR-path – visits all ext. vtcs – all comp. assigned to int. vtcs X Y Planar graph G P α Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P TSDR-path: Tutte path + System of Distinct Representatives: Injective assignment of comp. to attachment pts

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Tutte paths

Tint-path: –TSDR-path – visits all ext. vtcs – all comp. assigned to int. vtcs X Y Planar graph G P α Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P TSDR-path: Tutte path + System of Distinct Representatives: Injective assignment of comp. to attachment pts

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Tutte paths

Tint-path: –TSDR-path – visits all ext. vtcs – all comp. assigned to int. vtcs X Y Planar graph G P α Tutte path: Path from X to Y via α Every outer comp. attached to 2 vtcs of P Every comp. attached to ≤3 vtcs of P TSDR-path: Tutte path + System of Distinct Representatives: Injective assignment of comp. to attachment pts

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Chiba & Nishizeki ’89] G 4-conn. ⇒ Tutte path in O(n) time [Schmid & Schmidt ’18] . . . in O(n2) time (= Hamil. path) . . . in O(n) time [Schmid & Schmidt ’15] . . . in O(n2) time [Gao, Richter & Yu ’95, ’06] G 3-conn., X, Y, α on outer face ⇒ TSDR-path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

Tint

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Chiba & Nishizeki ’89] G 4-conn. ⇒ Tutte path in O(n) time [Schmid & Schmidt ’18] . . . in O(n2) time (= Hamil. path) . . . in O(n) time [Schmid & Schmidt ’15] . . . in O(n2) time [Gao, Richter & Yu ’95, ’06] G 3-conn., X, Y, α on outer face ⇒ TSDR-path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

Tint

int.

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What is known?

X Y α [Tutte ’77] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Thomassen ’83] G 2-conn., X, Y, α on outer face ⇒ Tutte path [Chiba & Nishizeki ’89] G 4-conn. ⇒ Tutte path in O(n) time [Schmid & Schmidt ’18] . . . in O(n2) time (= Hamil. path) . . . in O(n) time [Schmid & Schmidt ’15] . . . in O(n2) time [Gao, Richter & Yu ’95, ’06] G 3-conn., X, Y, α on outer face ⇒ TSDR-path [Sanders ’96] G 2-conn., X, Y, α on outer face ⇒ Tutte path

Tint

int.

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Triangulated Graphs

X Y α

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Triangulated Graphs

X Y α

[Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated Graphs

X Y α

[Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated Graphs

X Y α

[Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated Graphs

X Y α

[Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated Graphs

X Y α

triangulation ⇒ Tutte path in O(n) time. [Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated Graphs

X Y α

k vertices triangulation ⇒ Tutte path in O(n) time. [Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated Graphs

X Y α

k vertices 2k − 5 int. faces triangulation ⇒ Tutte path in O(n) time. [Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated Graphs

X Y α

k vertices 2k − 5 int. faces k − 3 int. vtcs triangulation ⇒ Tutte path in O(n) time. [Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated Graphs

X Y α

k vertices 2k − 5 int. faces k − 2 int. edges in P k − 3 int. vtcs triangulation ⇒ Tutte path in O(n) time. [Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated Graphs

X Y α

k vertices 2k − 5 int. faces k − 2 int. edges in P k − 3 int. vtcs triangulation ⇒ Tutte path in O(n) time. [Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated Graphs

X Y α

k vertices 2k − 5 int. faces k − 2 int. edges in P k − 3 int. vtcs triangulation ⇒ Tutte path in O(n) time. [Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Substitution Trick

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Substitution Trick

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Substitution Trick

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Substitution Trick

X Y α

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Substitution Trick

X Y α

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Substitution Trick

X Y α

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Substitution Trick

X Y α

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Substitution Trick

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Triangulated graphs

X Y α

k vertices 2k − 5 faces k − 2 edges in P−α k − 3 int. vtcs triangulation ⇒ Tutte path in O(n) time. [Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated graphs

X Y α

k vertices 2k − 5 faces k − 2 edges in P−α k − 3 int. vtcs triangulation ⇒ Tutte path in O(n) time. [Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated graphs

X Y α

k vertices 2k − 5 faces k − 2 edges in P−α k − 3 int. vtcs triangulation ⇒ Tutte path in O(n) time. [Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time.

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Triangulated graphs

X Y α

k vertices 2k − 5 faces k − 2 edges in P−α k − 3 int. vtcs triangulation ⇒ Tutte path in O(n) time. [Asano, Kikuchi & Saito ’85] 4-conn. triangulation ⇒ Hamiltonian path in O(n) time. Tint-

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Corner-3-connectivity

int. 3-conn.

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Corner-3-connectivity

int. 3-conn.

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Corner-3-connectivity

int. 3-conn.

corner-3-conn.

X Y α

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Corner-3-connectivity

int. 3-conn.

corner-3-conn.

X Y α

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Corner-3-connectivity

int. 3-conn.

corner-3-conn. side

X Y α

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Corner-3-connectivity

int. 3-conn.

corner-3-conn. side

X Y α

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Corner-3-connectivity

int. 3-conn.

corner-3-conn. side

X Y α

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Corner-3-connectivity

int. 3-conn.

corner-3-conn. side

X Y α

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Corner-3-connectivity

int. 3-conn.

corner-3-conn. side

X Y α

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Corner-3-connectivity

int. 3-conn.

corner-3-conn. side

X Y α

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Corner-3-connectivity

int. 3-conn.

corner-3-conn. side

X Y α

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Corner-3-connectivity

int. 3-conn.

corner-3-conn. side

X Y α

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Corner-3-connectivity

int. 3-conn.

corner-3-conn. side

X Y α

G is corner-3-conn., X, Y, α on outer face ⇒ Tint-path

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Corner-3-connectivity

int. 3-conn.

corner-3-conn. side

X Y α

G is corner-3-conn., X, Y, α on outer face ⇒ Tint-path

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Case 1: Outer Face is Triangle

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Case 1: Outer Face is Triangle

X Y α

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Case 1: Outer Face is Triangle

X Y α

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Case 1: Outer Face is Triangle

X Y α

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Case 2: left-right cutting pair

X Y α

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Case 2: left-right cutting pair

X Y α

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Case 2: left-right cutting pair

X Y Gt Gt α

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Case 2: left-right cutting pair

X Y Gb Gt Gt Gb α

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Case 2: left-right cutting pair

X Y Gb Gt Gt Gb α

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Case 2: left-right cutting pair

X Y Gb Gt X Y Gt Gb α α

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Case 2: left-right cutting pair

X Y Gb Gt X Y Gt Gb α α

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Case 2: left-right cutting pair

X Y Gb Gt X Y Gt Gb X Y α α α

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Case 2: left-right cutting pair

X Y Gb Gt X Y Gt Gb X Y α α α

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Case 2: left-right cutting pair

X Y Gb Gt X Y Gt Gb X Y α α α

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Case 3: top-right cutting pair

X Y α

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Case 3: top-right cutting pair

Gb X Y Gb α

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Case 3: top-right cutting pair

Gb Gt X Y Gb Gt α

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Case 3: top-right cutting pair

Gb Gt X Y Gb Gt α

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Case 3: top-right cutting pair

Gb Gt X Y Gb Y X Gt α α

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Case 3: top-right cutting pair

Gb Gt X Y Gb Y X Gt α α

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Case 3: top-right cutting pair

Gb Gt X Y Gb Y X Y Gt X α α α

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Case 3: top-right cutting pair

Gb Gt X Y Gb Y X Y Gt X α α α

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Case 3: top-right cutting pair

Gb Gt X Y Gb Y X Y Gt X α α α

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Case 3: top-right cutting pair

Gb Gt X Y Gb Y X Y Gt X α α α

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Case 3: top-right cutting pair

Gb Gt X Y Gb Y X Y Gt X α α α

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Case 3’: top-left cutting pair

Gb Gt X Y Gb Y X Y Gt X α α α y

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Case 3”: top-bottom cutting pair

Gb Gt X Y Y Gt X Gb X Y α α α

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Case 4: No cutting pair

X Y YX α

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Case 4: No cutting pair

X Y Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs YX α

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Case 4: No cutting pair

X Y Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs xs

=x0

YX α

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Case 4: No cutting pair

X Y x1 x2 xi Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs fi xs

=x0

YX α

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Case 4: No cutting pair

X Y x1 x2 xi Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side fi xs

=x0

YX α

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Case 4: No cutting pair

X Y x1 x2 fi xi Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side xs

=x0

YX α

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Case 4: No cutting pair

X Y x1 x2 fi xi Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side xs

=x0

YX α G0

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Case 4: No cutting pair

X Y x1 x2 fi xi Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side G1 xs

=x0

YX α G0

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Case 4: No cutting pair

X Y x1 x2 fi xi Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side G2 G1 xs

=x0

YX α G0

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Case 4: No cutting pair

X Y x1 x2 fi xi Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side G2 G1 xs

=x0

YX α G0 Gi

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Case 4: No cutting pair

X Y x1 x2 fi xi Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side G2 G1 xs

=x0

YX α G0 Gi Gs

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Case 4: No cutting pair

X Y YX G0 G2 Gi Gs α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

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Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Gs α α G0 Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

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Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Gs α α G0 Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

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Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Gs α α G0 Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

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Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Gs α α G0 Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

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Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Gs α α G0 Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

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Case 4: No cutting pair

X Y YX G0 G2 Gi Gs α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

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Case 4: No cutting pair

X Y YX G0 G2 Gi Gs α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

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Case 4: No cutting pair

X Y YX G0 G2 Gi Gs α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

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Case 4: No cutting pair

X Y YX G0 G2 Gi Gs α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

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Case 4: No cutting pair

X Y YX G0 G2 Gi Gs α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

SLIDE 130

Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Gs α α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

SLIDE 131

Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Gs α α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

SLIDE 132

Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Gs α α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

SLIDE 133

Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Gs α α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

SLIDE 134

Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Gs α α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

SLIDE 135

Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Gs α α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

SLIDE 136

Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Y X Gs α α α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

SLIDE 137

Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Y X Gs α α α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

SLIDE 138

Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Y X Gs α α α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

SLIDE 139

Case 4: No cutting pair

X Y YX G0 G2 Gi X Y Y X Gs C2 α α α Necklace YX=x0, f1, x1, . . . , xs−1, fs, xs,xi face-adj. to right side,G1 = ∅

SLIDE 140

Running Time

SLIDE 141

Running Time

SLIDE 142

Running Time

Store:

SLIDE 143

Running Time

Store:

corners

SLIDE 144

Running Time

Store:

faces: all vtcs on each side
corners

SLIDE 145

Running Time

Store:

faces: all vtcs on each side
vtcs: all face-incidences to each side
corners

SLIDE 146

Running Time

Store:

faces: all vtcs on each side
vtcs: all face-incidences to each side
corners
sides: all cutting pairs

SLIDE 147

Necklace scan

X Y YX α

SLIDE 148

Necklace scan

X Y YX α

SLIDE 149

Necklace scan

X Y YX α

SLIDE 150

Necklace scan

X Y YX α

SLIDE 151

Necklace scan

X Y YX α

SLIDE 152

Necklace scan

X Y YX α

SLIDE 153

Necklace scan

X Y YX α

SLIDE 154

Necklace scan

X Y YX α

SLIDE 155

Necklace scan

X Y YX α

SLIDE 156

Necklace scan

X Y YX α

SLIDE 157

Necklace scan

X Y YX α

SLIDE 158

Necklace scan

face gets scanned: X Y YX α

SLIDE 159

Necklace scan

face gets scanned: X Y YX α

SLIDE 160

Necklace scan

face gets scanned:

⇒ one vtx becomes outer

X Y YX α

SLIDE 161

Necklace scan

face gets scanned:

⇒ one vtx becomes outer

X Y YX α

SLIDE 162

Necklace scan

face gets scanned:

⇒ one vtx becomes outer ⇒ O(1) times

X Y YX α

SLIDE 163

Necklace scan

face gets scanned:

⇒ one vtx becomes outer ⇒ O(1) times ⇒ O(∑ f deg( f )) = O(n) time

X Y YX α

SLIDE 164

Necklace scan

face gets scanned:

⇒ one vtx becomes outer ⇒ O(1) times ⇒ O(∑ f deg( f )) = O(n) time

Theorem.

G int. 3-conn., X, Y, α on outer face ⇒ Tint-path in O(n) time. X Y YX α

SLIDE 165

Applications

SLIDE 166

Applications

X Y α

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time.

SLIDE 167

Applications

X Y α

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time.

SLIDE 168

Applications

X Y α

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time.

SLIDE 169

Applications

W X Y U

Theorem.

G int. 3-conn. ⇒ 2-circuit in O(n) time.

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time. X Y α

SLIDE 170

Applications

W X Y U

Theorem.

G int. 3-conn. ⇒ 2-circuit in O(n) time.

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time. X Y α

SLIDE 171

Applications

W X Y U

Theorem.

G int. 3-conn. ⇒ 2-circuit in O(n) time.

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time. X Y α

SLIDE 172

Applications

W X Y U

Theorem.

G int. 3-conn. ⇒ 2-circuit in O(n) time.

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time. X Y α

SLIDE 173

Applications

W X Y U

Theorem.

G int. 3-conn. ⇒ 2-circuit in O(n) time.

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time. X Y α

SLIDE 174

Conclusion

SLIDE 175

Conclusion

Theorem.

G int. 3-conn., X, Y, α on outer face ⇒ Tint-path in O(n) time.

SLIDE 176

Conclusion

Theorem. Theorem.

G 2-conn., X, Y, α on outer face ⇒ Tutte path in O(n) time. G int. 3-conn., X, Y, α on outer face ⇒ Tint-path in O(n) time.

SLIDE 177

Conclusion

Theorem. Theorem.

G 2-conn., X, Y, α on outer face ⇒ Tutte path in O(n) time.

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time. G int. 3-conn., X, Y, α on outer face ⇒ Tint-path in O(n) time.

SLIDE 178

Conclusion

Theorem. Theorem.

G 2-conn., X, Y, α on outer face ⇒ Tutte path in O(n) time.

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time.

Theorem.

G int. 3-conn. ⇒ 2-circuit in O(n) time. G int. 3-conn., X, Y, α on outer face ⇒ Tint-path in O(n) time.

SLIDE 179

Conclusion

Theorem. Theorem.

G 2-conn., X, Y, α on outer face ⇒ Tutte path in O(n) time.

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time.

Theorem.

G int. 3-conn. ⇒ 2-circuit in O(n) time. X, Y, α on different faces? G int. 3-conn., X, Y, α on outer face ⇒ Tint-path in O(n) time.

SLIDE 180

Conclusion

Theorem. Theorem.

G 2-conn., X, Y, α on outer face ⇒ Tutte path in O(n) time.

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time.

Theorem.

G int. 3-conn. ⇒ 2-circuit in O(n) time. X, Y, α on different faces? Non-planar graphs? G int. 3-conn., X, Y, α on outer face ⇒ Tint-path in O(n) time.

SLIDE 181

Conclusion

Theorem. Theorem.

G 2-conn., X, Y, α on outer face ⇒ Tutte path in O(n) time.

Theorem.

G int. 3-conn. ⇒ binary spanning tree in O(n) time.

Theorem.

G int. 3-conn. ⇒ 2-circuit in O(n) time. X, Y, α on different faces? Non-planar graphs? G int. 3-conn., X, Y, α on outer face ⇒ Tint-path in O(n) time.