Algorithmic Graph Theory (SS2016) Chapter 3 Simple Intersection-Graphs Walter Unger
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(2) Intersection-Graphs 3. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. Inhaltsverzeichnis. Walter Unger 21.12.2018 12:38. Contents I. 1. Intersection-Graphs Basics Problems. 2. Interval-graphs Introduction Colouring Independent Sets and Cliques. 3. Permutation-Graphs Introduction Colouring. 4. Arc-Graphs. Introduction Colouring. 5. Circle-Grahs Introduction Colouring Construction Colourings Independent Sets and Cliques. 6. Concluding Remarks Segment-graphs Disk-graphs Overview. Z. Concluding Remarks SS2016.
(3) Intersection-Graphs 3:1. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. 1/8. Circle-Grahs Walter Unger 21.12.2018 12:38. Basics A graph consists of nodes, which are “connected” by some relation. Often we have objects, for which some relation exists. Possible relations: Objects Objects Objects Objects. have some common property. are neighbours. have some limited distance. intersect.. We define intersection-graphs using the later relation.. Z. Concluding Remarks SS2016.
(4) Intersection-Graphs 3:1. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. 2/8. Circle-Grahs Walter Unger 21.12.2018 12:38. Basics A graph consists of nodes, which are “connected” by some relation. Often we have objects, for which some relation exists. Possible relations: Objects Objects Objects Objects. have some common property. are neighbours. have some limited distance. intersect.. We define intersection-graphs using the later relation.. Z. Concluding Remarks SS2016.
(5) Intersection-Graphs 3:1. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. 3/8. Circle-Grahs Walter Unger 21.12.2018 12:38. Basics A graph consists of nodes, which are “connected” by some relation. Often we have objects, for which some relation exists. Possible relations: Objects Objects Objects Objects. have some common property. are neighbours. have some limited distance. intersect.. We define intersection-graphs using the later relation.. Z. Concluding Remarks SS2016.
(6) Intersection-Graphs 3:1. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. 4/8. Circle-Grahs Walter Unger 21.12.2018 12:38. Basics A graph consists of nodes, which are “connected” by some relation. Often we have objects, for which some relation exists. Possible relations: Objects Objects Objects Objects. have some common property. are neighbours. have some limited distance. intersect.. We define intersection-graphs using the later relation.. Z. Concluding Remarks SS2016.
(7) Intersection-Graphs 3:1. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. 5/8. Circle-Grahs Walter Unger 21.12.2018 12:38. Basics A graph consists of nodes, which are “connected” by some relation. Often we have objects, for which some relation exists. Possible relations: Objects Objects Objects Objects. have some common property. are neighbours. have some limited distance. intersect.. We define intersection-graphs using the later relation.. Z. Concluding Remarks SS2016.
(8) Intersection-Graphs 3:1. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. 6/8. Circle-Grahs Walter Unger 21.12.2018 12:38. Basics A graph consists of nodes, which are “connected” by some relation. Often we have objects, for which some relation exists. Possible relations: Objects Objects Objects Objects. have some common property. are neighbours. have some limited distance. intersect.. We define intersection-graphs using the later relation.. Z. Concluding Remarks SS2016.
(9) Intersection-Graphs 3:1. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. 7/8. Circle-Grahs Walter Unger 21.12.2018 12:38. Basics A graph consists of nodes, which are “connected” by some relation. Often we have objects, for which some relation exists. Possible relations: Objects Objects Objects Objects. have some common property. are neighbours. have some limited distance. intersect.. We define intersection-graphs using the later relation.. Z. Concluding Remarks SS2016.
(10) Intersection-Graphs 3:1. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. 8/8. Circle-Grahs Walter Unger 21.12.2018 12:38. Basics A graph consists of nodes, which are “connected” by some relation. Often we have objects, for which some relation exists. Possible relations: Objects Objects Objects Objects. have some common property. are neighbours. have some limited distance. intersect.. We define intersection-graphs using the later relation.. Z. Concluding Remarks SS2016.
(11) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 1/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(12) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 2/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(13) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 3/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(14) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 4/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(15) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 5/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(16) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 6/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(17) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 7/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(18) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 8/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(19) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 9/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(20) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 10/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(21) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 11/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(22) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 12/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(23) Intersection-Graphs 3:2. Basics. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 13/13. Walter Unger 21.12.2018 12:38. Definition Definition A graph G = (V , E ) is called intersection-graph of a set M of objects, iff G = (V , E ) is isomorphic to H = (M, {{a, b} | a ∩ b 6= ∅}). M is called the intersection representation of G . Possible families of objects are: Intervals on a line. Arc of a circle. Chords of a circle. Circles in the plane. Parallelograms between two lines. And lots more.. By using different classes of object we get different graph classes.. Z. Concluding Remarks SS2016.
(24) Intersection-Graphs 3:3. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. 1/7. Colouring Definition A graph G = (V , E ) is k-colourable iff: ∃f : V 7→ {1, ..., k} : ∀(a, b) ∈ E , f (a) 6= f (b). The function f is called colouring of G . Definition χ(G ) is the chromatic number χ(G ) of G , iff G is χ(G )-colourable, but is not (χ(G ) − 1)-colourable.. Circle-Grahs Walter Unger 21.12.2018 12:38. Z. Concluding Remarks SS2016.
(25) Intersection-Graphs 3:3. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. 2/7. Colouring Definition A graph G = (V , E ) is k-colourable iff: ∃f : V 7→ {1, ..., k} : ∀(a, b) ∈ E , f (a) 6= f (b). The function f is called colouring of G . Definition χ(G ) is the chromatic number χ(G ) of G , iff G is χ(G )-colourable, but is not (χ(G ) − 1)-colourable.. Circle-Grahs Walter Unger 21.12.2018 12:38. Z. Concluding Remarks SS2016.
(26) Intersection-Graphs 3:3. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. 3/7. Colouring Definition A graph G = (V , E ) is k-colourable iff: ∃f : V 7→ {1, ..., k} : ∀(a, b) ∈ E , f (a) 6= f (b). The function f is called colouring of G . Definition χ(G ) is the chromatic number χ(G ) of G , iff G is χ(G )-colourable, but is not (χ(G ) − 1)-colourable.. Circle-Grahs Walter Unger 21.12.2018 12:38. Z. Concluding Remarks SS2016.
(27) Intersection-Graphs 3:3. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. 4/7. Colouring Definition A graph G = (V , E ) is k-colourable iff: ∃f : V 7→ {1, ..., k} : ∀(a, b) ∈ E , f (a) 6= f (b). The function f is called colouring of G . Definition χ(G ) is the chromatic number χ(G ) of G , iff G is χ(G )-colourable, but is not (χ(G ) − 1)-colourable.. Circle-Grahs Walter Unger 21.12.2018 12:38. Z. Concluding Remarks SS2016.
(28) Intersection-Graphs 3:3. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. 5/7. Colouring Definition A graph G = (V , E ) is k-colourable iff: ∃f : V 7→ {1, ..., k} : ∀(a, b) ∈ E , f (a) 6= f (b). The function f is called colouring of G . Definition χ(G ) is the chromatic number χ(G ) of G , iff G is χ(G )-colourable, but is not (χ(G ) − 1)-colourable.. Circle-Grahs Walter Unger 21.12.2018 12:38. Z. Concluding Remarks SS2016.
(29) Intersection-Graphs 3:3. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. 6/7. Colouring Definition A graph G = (V , E ) is k-colourable iff: ∃f : V 7→ {1, ..., k} : ∀(a, b) ∈ E , f (a) 6= f (b). The function f is called colouring of G . Definition χ(G ) is the chromatic number χ(G ) of G , iff G is χ(G )-colourable, but is not (χ(G ) − 1)-colourable.. Circle-Grahs Walter Unger 21.12.2018 12:38. Z. Concluding Remarks SS2016.
(30) Intersection-Graphs 3:3. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. 7/7. Colouring Definition A graph G = (V , E ) is k-colourable iff: ∃f : V 7→ {1, ..., k} : ∀(a, b) ∈ E , f (a) 6= f (b). The function f is called colouring of G . Definition χ(G ) is the chromatic number χ(G ) of G , iff G is χ(G )-colourable, but is not (χ(G ) − 1)-colourable.. Circle-Grahs Walter Unger 21.12.2018 12:38. Z. Concluding Remarks SS2016.
(31) Intersection-Graphs 3:4. Problems. Interval-graphs. Perm.-Gr.. 1/3. Colouring Problems Definition The graph-to-colour problem is the following: Input: G a graph Output: Optimal colouring of G . Definition The colouring problem is the following: Input: k ∈ N and a graph G Output: Is G k-colourable? Definition The k-colouring problem is the following: Input: G a Graph Output: Is G k-colourable?. Arc-Graphs. Circle-Grahs Walter Unger 21.12.2018 12:38. Z. Concluding Remarks SS2016.
(32) Intersection-Graphs 3:4. Problems. Interval-graphs. Perm.-Gr.. 2/3. Colouring Problems Definition The graph-to-colour problem is the following: Input: G a graph Output: Optimal colouring of G . Definition The colouring problem is the following: Input: k ∈ N and a graph G Output: Is G k-colourable? Definition The k-colouring problem is the following: Input: G a Graph Output: Is G k-colourable?. Arc-Graphs. Circle-Grahs Walter Unger 21.12.2018 12:38. Z. Concluding Remarks SS2016.
(33) Intersection-Graphs 3:4. Problems. Interval-graphs. Perm.-Gr.. 3/3. Colouring Problems Definition The graph-to-colour problem is the following: Input: G a graph Output: Optimal colouring of G . Definition The colouring problem is the following: Input: k ∈ N and a graph G Output: Is G k-colourable? Definition The k-colouring problem is the following: Input: G a Graph Output: Is G k-colourable?. Arc-Graphs. Circle-Grahs Walter Unger 21.12.2018 12:38. Z. Concluding Remarks SS2016.
(34) Intersection-Graphs 3:5. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 1/6. Walter Unger 21.12.2018 12:38. Independent Set Definition A graph G = (V , E ) contains an independent set of size k, iff ∃S ⊂ V : |S| = k ∧ ∀a, b ∈ S, a 6= b : (a, b) 6∈ E . Definition α(G ) denotes the size of the largest independent set: G contains an independet set of size α(G ), but no independet set of size α(G ) + 1.. Z. Concluding Remarks SS2016.
(35) Intersection-Graphs 3:5. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 2/6. Walter Unger 21.12.2018 12:38. Independent Set Definition A graph G = (V , E ) contains an independent set of size k, iff ∃S ⊂ V : |S| = k ∧ ∀a, b ∈ S, a 6= b : (a, b) 6∈ E . Definition α(G ) denotes the size of the largest independent set: G contains an independet set of size α(G ), but no independet set of size α(G ) + 1.. Z. Concluding Remarks SS2016.
(36) Intersection-Graphs 3:5. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 3/6. Walter Unger 21.12.2018 12:38. Independent Set Definition A graph G = (V , E ) contains an independent set of size k, iff ∃S ⊂ V : |S| = k ∧ ∀a, b ∈ S, a 6= b : (a, b) 6∈ E . Definition α(G ) denotes the size of the largest independent set: G contains an independet set of size α(G ), but no independet set of size α(G ) + 1.. Z. Concluding Remarks SS2016.
(37) Intersection-Graphs 3:5. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 4/6. Walter Unger 21.12.2018 12:38. Independent Set Definition A graph G = (V , E ) contains an independent set of size k, iff ∃S ⊂ V : |S| = k ∧ ∀a, b ∈ S, a 6= b : (a, b) 6∈ E . Definition α(G ) denotes the size of the largest independent set: G contains an independet set of size α(G ), but no independet set of size α(G ) + 1.. Z. Concluding Remarks SS2016.
(38) Intersection-Graphs 3:5. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 5/6. Walter Unger 21.12.2018 12:38. Independent Set Definition A graph G = (V , E ) contains an independent set of size k, iff ∃S ⊂ V : |S| = k ∧ ∀a, b ∈ S, a 6= b : (a, b) 6∈ E . Definition α(G ) denotes the size of the largest independent set: G contains an independet set of size α(G ), but no independet set of size α(G ) + 1.. Z. Concluding Remarks SS2016.
(39) Intersection-Graphs 3:5. Problems. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 6/6. Walter Unger 21.12.2018 12:38. Independent Set Definition A graph G = (V , E ) contains an independent set of size k, iff ∃S ⊂ V : |S| = k ∧ ∀a, b ∈ S, a 6= b : (a, b) 6∈ E . Definition α(G ) denotes the size of the largest independent set: G contains an independet set of size α(G ), but no independet set of size α(G ) + 1.. Z. Concluding Remarks SS2016.
(40) Intersection-Graphs 3:6. Interval-graphs. Perm.-Gr.. Problems. Arc-Graphs. Circle-Grahs Walter Unger 21.12.2018 12:38. Definitions Definition Let G = (V , E ) be a graph. α(G ) ω(G ) χ(G ). = = =. χ(G ). =. More notations: ω(G ) = α(G ), α(G ) = ω(G ) = β0 (G ), κ(G ) = χ(G ). max{ |V 0 | ; V 0 ⊂ V ∧ ∀a, b ∈ V 0 : (a, b) 6∈ E } max{ |V 0 | ; V 0 ⊂ V ∧ ∀a, b ∈ V 0 : (a, b) ∈ E } min{ k ; ∃V1 , V2 , . . . , Vk : ∪ki=1 Vi = V ∧ ∀i : 1 6 i 6 k : ∀a, b ∈ Vi : (a, b) 6∈ E } min{ k ; ∃V1 , V2 , . . . , Vk : ∪ki=1 Vi = V ∧ ∀i : 1 6 i 6 k : ∀a, b ∈ Vi : (a, b) ∈ E }. Z. Concluding Remarks SS2016.
(41) Intersection-Graphs 3:7. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 1/11. Walter Unger 21.12.2018 12:38. First simple Example Time of activity of a register (construction of a compiler) Program secments: · · · Read(A) · · · Write(B) · · · Living time of a variable A: Maximal interval Starting with a Write(A). Ending by the last Read(A). Such that no further Write(A) is between this two points.. Problem: how manny registers are needed? D.h. assign for each living time of a variable a register. Example: (0, 10), (3, 7), (9, 20), (25, 50), (12, 34), (6, 16), (17, 26) (11, 46), (23, 26), (30, 46), (19, 27) 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(42) Intersection-Graphs 3:7. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 2/11. Walter Unger 21.12.2018 12:38. First simple Example Time of activity of a register (construction of a compiler) Program secments: · · · Read(A) · · · Write(B) · · · Living time of a variable A: Maximal interval Starting with a Write(A). Ending by the last Read(A). Such that no further Write(A) is between this two points.. Problem: how manny registers are needed? D.h. assign for each living time of a variable a register. Example: (0, 10), (3, 7), (9, 20), (25, 50), (12, 34), (6, 16), (17, 26) (11, 46), (23, 26), (30, 46), (19, 27) 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(43) Intersection-Graphs 3:7. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 3/11. Walter Unger 21.12.2018 12:38. First simple Example Time of activity of a register (construction of a compiler) Program secments: · · · Read(A) · · · Write(B) · · · Living time of a variable A: Maximal interval Starting with a Write(A). Ending by the last Read(A). Such that no further Write(A) is between this two points.. Problem: how manny registers are needed? D.h. assign for each living time of a variable a register. Example: (0, 10), (3, 7), (9, 20), (25, 50), (12, 34), (6, 16), (17, 26) (11, 46), (23, 26), (30, 46), (19, 27) 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(44) Intersection-Graphs 3:7. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 4/11. Walter Unger 21.12.2018 12:38. First simple Example Time of activity of a register (construction of a compiler) Program secments: · · · Read(A) · · · Write(B) · · · Living time of a variable A: Maximal interval Starting with a Write(A). Ending by the last Read(A). Such that no further Write(A) is between this two points.. Problem: how manny registers are needed? D.h. assign for each living time of a variable a register. Example: (0, 10), (3, 7), (9, 20), (25, 50), (12, 34), (6, 16), (17, 26) (11, 46), (23, 26), (30, 46), (19, 27) 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(45) Intersection-Graphs 3:7. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 5/11. Walter Unger 21.12.2018 12:38. First simple Example Time of activity of a register (construction of a compiler) Program secments: · · · Read(A) · · · Write(B) · · · Living time of a variable A: Maximal interval Starting with a Write(A). Ending by the last Read(A). Such that no further Write(A) is between this two points.. Problem: how manny registers are needed? D.h. assign for each living time of a variable a register. Example: (0, 10), (3, 7), (9, 20), (25, 50), (12, 34), (6, 16), (17, 26) (11, 46), (23, 26), (30, 46), (19, 27) 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(46) Intersection-Graphs 3:7. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 6/11. Walter Unger 21.12.2018 12:38. First simple Example Time of activity of a register (construction of a compiler) Program secments: · · · Read(A) · · · Write(B) · · · Living time of a variable A: Maximal interval Starting with a Write(A). Ending by the last Read(A). Such that no further Write(A) is between this two points.. Problem: how manny registers are needed? D.h. assign for each living time of a variable a register. Example: (0, 10), (3, 7), (9, 20), (25, 50), (12, 34), (6, 16), (17, 26) (11, 46), (23, 26), (30, 46), (19, 27) 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(47) Intersection-Graphs 3:7. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 7/11. Walter Unger 21.12.2018 12:38. First simple Example Time of activity of a register (construction of a compiler) Program secments: · · · Read(A) · · · Write(B) · · · Living time of a variable A: Maximal interval Starting with a Write(A). Ending by the last Read(A). Such that no further Write(A) is between this two points.. Problem: how manny registers are needed? D.h. assign for each living time of a variable a register. Example: (0, 10), (3, 7), (9, 20), (25, 50), (12, 34), (6, 16), (17, 26) (11, 46), (23, 26), (30, 46), (19, 27) 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(48) Intersection-Graphs 3:7. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 8/11. Walter Unger 21.12.2018 12:38. First simple Example Time of activity of a register (construction of a compiler) Program secments: · · · Read(A) · · · Write(B) · · · Living time of a variable A: Maximal interval Starting with a Write(A). Ending by the last Read(A). Such that no further Write(A) is between this two points.. Problem: how manny registers are needed? D.h. assign for each living time of a variable a register. Example: (0, 10), (3, 7), (9, 20), (25, 50), (12, 34), (6, 16), (17, 26) (11, 46), (23, 26), (30, 46), (19, 27) 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(49) Intersection-Graphs 3:7. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 9/11. Walter Unger 21.12.2018 12:38. First simple Example Time of activity of a register (construction of a compiler) Program secments: · · · Read(A) · · · Write(B) · · · Living time of a variable A: Maximal interval Starting with a Write(A). Ending by the last Read(A). Such that no further Write(A) is between this two points.. Problem: how manny registers are needed? D.h. assign for each living time of a variable a register. Example: (0, 10), (3, 7), (9, 20), (25, 50), (12, 34), (6, 16), (17, 26) (11, 46), (23, 26), (30, 46), (19, 27) 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(50) Intersection-Graphs 3:7. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 10/11. Walter Unger 21.12.2018 12:38. First simple Example Time of activity of a register (construction of a compiler) Program secments: · · · Read(A) · · · Write(B) · · · Living time of a variable A: Maximal interval Starting with a Write(A). Ending by the last Read(A). Such that no further Write(A) is between this two points.. Problem: how manny registers are needed? D.h. assign for each living time of a variable a register. Example: (0, 10), (3, 7), (9, 20), (25, 50), (12, 34), (6, 16), (17, 26) (11, 46), (23, 26), (30, 46), (19, 27) 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(51) Intersection-Graphs 3:7. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 11/11. Walter Unger 21.12.2018 12:38. First simple Example Time of activity of a register (construction of a compiler) Program secments: · · · Read(A) · · · Write(B) · · · Living time of a variable A: Maximal interval Starting with a Write(A). Ending by the last Read(A). Such that no further Write(A) is between this two points.. Problem: how manny registers are needed? D.h. assign for each living time of a variable a register. Example: (0, 10), (3, 7), (9, 20), (25, 50), (12, 34), (6, 16), (17, 26) (11, 46), (23, 26), (30, 46), (19, 27) 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(52) Intersection-Graphs 3:8. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. 1/7. Circle-Grahs Walter Unger 21.12.2018 12:38. Interval-graphs Definition (Interval-graphs) A graph G = (V , E ) is called intervall-graph, iff it is the intersection graph of a set of intervals on a line. An interval-graph is called proper, iff no interval is contained in an other interval. 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. f. g e d. c b. a 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 g. e. r c. a. j i. h. b. d. Z. Concluding Remarks SS2016.
(53) Intersection-Graphs 3:8. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. 2/7. Circle-Grahs Walter Unger 21.12.2018 12:38. Interval-graphs Definition (Interval-graphs) A graph G = (V , E ) is called intervall-graph, iff it is the intersection graph of a set of intervals on a line. An interval-graph is called proper, iff no interval is contained in an other interval. 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. f. g e d. c b. a 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 g. e. r c. a. j i. h. b. d. Z. Concluding Remarks SS2016.
(54) Intersection-Graphs 3:8. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. 3/7. Circle-Grahs Walter Unger 21.12.2018 12:38. Interval-graphs Definition (Interval-graphs) A graph G = (V , E ) is called intervall-graph, iff it is the intersection graph of a set of intervals on a line. An interval-graph is called proper, iff no interval is contained in an other interval. 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. f. g e d. c b. a 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 g. e. r c. a. j i. h. b. d. Z. Concluding Remarks SS2016.
(55) Intersection-Graphs 3:8. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. 4/7. Circle-Grahs Walter Unger 21.12.2018 12:38. Interval-graphs Definition (Interval-graphs) A graph G = (V , E ) is called intervall-graph, iff it is the intersection graph of a set of intervals on a line. An interval-graph is called proper, iff no interval is contained in an other interval. 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. f. g e d. c b. a 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 g. e. r c. a. j i. h. b. d. Z. Concluding Remarks SS2016.
(56) Intersection-Graphs 3:8. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. 5/7. Circle-Grahs Walter Unger 21.12.2018 12:38. Interval-graphs Definition (Interval-graphs) A graph G = (V , E ) is called intervall-graph, iff it is the intersection graph of a set of intervals on a line. An interval-graph is called proper, iff no interval is contained in an other interval. 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. f. g e d. c b. a 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 g. e. r c. a. j i. h. b. d. Z. Concluding Remarks SS2016.
(57) Intersection-Graphs 3:8. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. 6/7. Circle-Grahs Walter Unger 21.12.2018 12:38. Interval-graphs Definition (Interval-graphs) A graph G = (V , E ) is called intervall-graph, iff it is the intersection graph of a set of intervals on a line. An interval-graph is called proper, iff no interval is contained in an other interval. 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. f. g e d. c b. a 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 g. e. r c. a. j i. h. b. d. Z. Concluding Remarks SS2016.
(58) Intersection-Graphs 3:8. Introduction. Interval-graphs. Perm.-Gr.. Arc-Graphs. 7/7. Circle-Grahs Walter Unger 21.12.2018 12:38. Interval-graphs Definition (Interval-graphs) A graph G = (V , E ) is called intervall-graph, iff it is the intersection graph of a set of intervals on a line. An interval-graph is called proper, iff no interval is contained in an other interval. 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. f. g e d. c b. a 0. 2. 4. 6. 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 g. e. r c. a. j i. h. b. d. Z. Concluding Remarks SS2016.
(59) Intersection-Graphs 3:9. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 1/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: look for independent sets. 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(60) Intersection-Graphs 3:9. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 2/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: look for independent sets. 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(61) Intersection-Graphs 3:9. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 3/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: look for independent sets. 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(62) Intersection-Graphs 3:9. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 4/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: look for independent sets. 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(63) Intersection-Graphs 3:9. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 5/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: look for independent sets. 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(64) Intersection-Graphs 3:9. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 6/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: look for independent sets. 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(65) Intersection-Graphs 3:9. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 7/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: look for independent sets. 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f c a. b. g e d. Z. Concluding Remarks SS2016.
(66) Intersection-Graphs 3:9. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 8/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: look for independent sets. 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f. g e d. c b. a 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50. c a. b. d. Z. Concluding Remarks SS2016.
(67) Intersection-Graphs 3:9. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 9/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: look for independent sets. 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f. g e d. c b. a 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50. c a. b. d. Z. Concluding Remarks SS2016.
(68) Intersection-Graphs 3:9. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 10/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: look for independent sets. 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f. g e d. c b. a 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50. c a. b. d. Z. Concluding Remarks SS2016.
(69) Intersection-Graphs 3:9. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 11/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: look for independent sets. 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 k. j i. h f. g e d. c b. a 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50. c a. b. d. Z. Concluding Remarks SS2016.
(70) Intersection-Graphs 3:10. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 1/12. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: check the intervalls from left to the right (sorted by the left endpoints): 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h. k. i. e. g. c f. a. b. d. j. Z. Concluding Remarks SS2016.
(71) Intersection-Graphs 3:10. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 2/12. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: check the intervalls from left to the right (sorted by the left endpoints): 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h. k. i. e. g. c f. a. b. d. j. Z. Concluding Remarks SS2016.
(72) Intersection-Graphs 3:10. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 3/12. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: check the intervalls from left to the right (sorted by the left endpoints): 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h. k. i. e. g. c f. a. b. d. j. Z. Concluding Remarks SS2016.
(73) Intersection-Graphs 3:10. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 4/12. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: check the intervalls from left to the right (sorted by the left endpoints): 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h. k. i. e. g. c f. a. b. d. j. Z. Concluding Remarks SS2016.
(74) Intersection-Graphs 3:10. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 5/12. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: check the intervalls from left to the right (sorted by the left endpoints): 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h. k. i. e. g. c f. a. b. d. j. Z. Concluding Remarks SS2016.
(75) Intersection-Graphs 3:10. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 6/12. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: check the intervalls from left to the right (sorted by the left endpoints): 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h. k. i. e. g. c f. a. b. d. j. Z. Concluding Remarks SS2016.
(76) Intersection-Graphs 3:10. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 7/12. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: check the intervalls from left to the right (sorted by the left endpoints): 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h. k. i. e. g. c f. a. b. d. j. Z. Concluding Remarks SS2016.
(77) Intersection-Graphs 3:10. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 8/12. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: check the intervalls from left to the right (sorted by the left endpoints): 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h. k. i. e. g. c f. a. b. d. j. Z. Concluding Remarks SS2016.
(78) Intersection-Graphs 3:10. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 9/12. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: check the intervalls from left to the right (sorted by the left endpoints): 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h. k. i. e. g. c f. a. b. d. j. Z. Concluding Remarks SS2016.
(79) Intersection-Graphs 3:10. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 10/12. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: check the intervalls from left to the right (sorted by the left endpoints): 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h. k. i. e. g. c f. a. b. d. j. Z. Concluding Remarks SS2016.
(80) Intersection-Graphs 3:10. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 11/12. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: check the intervalls from left to the right (sorted by the left endpoints): 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h. k. i. e. g. c f. a. b. d. j. Z. Concluding Remarks SS2016.
(81) Intersection-Graphs 3:10. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 12/12. Walter Unger 21.12.2018 12:38. Model and Colouring (Idea) Idea: check the intervalls from left to the right (sorted by the left endpoints): 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h. k. i. e. g. c f. a. b. d. j. Z. Concluding Remarks SS2016.
(82) Intersection-Graphs 3:11. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 1/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Invariant) Determine the invariant: 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 j f. c a. b. d. e. h g i. k. Z. Concluding Remarks SS2016.
(83) Intersection-Graphs 3:11. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 2/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Invariant) Determine the invariant: 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 j f. c a. b. d. e. h g i. k. Z. Concluding Remarks SS2016.
(84) Intersection-Graphs 3:11. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 3/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Invariant) Determine the invariant: 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 j f. c a. b. d. e. h g i. k. Z. Concluding Remarks SS2016.
(85) Intersection-Graphs 3:11. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 4/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Invariant) Determine the invariant: 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 j f. c a. b. d. e. h g i. k. Z. Concluding Remarks SS2016.
(86) Intersection-Graphs 3:11. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 5/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Invariant) Determine the invariant: 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 j f. c a. b. d. e. h g i. k. Z. Concluding Remarks SS2016.
(87) Intersection-Graphs 3:11. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 6/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Invariant) Determine the invariant: 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 j f. c a. b. d. e. h g i. k. Z. Concluding Remarks SS2016.
(88) Intersection-Graphs 3:11. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 7/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Invariant) Determine the invariant: 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 j f. c a. b. d. e. h g i. k. Z. Concluding Remarks SS2016.
(89) Intersection-Graphs 3:11. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 8/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Invariant) Determine the invariant: 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 j f. c a. b. d. e. h g i. k. Z. Concluding Remarks SS2016.
(90) Intersection-Graphs 3:11. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 9/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Invariant) Determine the invariant: 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 j f. c a. b. d. e. h g i. k. Z. Concluding Remarks SS2016.
(91) Intersection-Graphs 3:11. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 10/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Invariant) Determine the invariant: 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 j f. c a. b. d. e. h g i. k. Z. Concluding Remarks SS2016.
(92) Intersection-Graphs 3:11. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 11/11. Walter Unger 21.12.2018 12:38. Model and Colouring (Invariant) Determine the invariant: 0. 2. 4. 6. 8. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 j f. c a. b. d. e. h g i. k. Z. Concluding Remarks SS2016.
(93) Intersection-Graphs 3:12. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 1/8. Walter Unger 21.12.2018 12:38. Colouring of Interval-graphs (Algorithm) Theorem The graph-to-colour problem is for interval-graphs in time O(n log(n)) solvable. 1. Sort the intervals by their left endpoints.. 2. Check all endpoints e from the left to the right.. 3. If e is the starting point of an interval, colour it with the smallest free colour.. 4. If e is the ending point of an interval I is, free the colour of I .. Invariant If a node v is coloured with colour k, then v is part of a k-clique.. Z. Concluding Remarks SS2016.
(94) Intersection-Graphs 3:12. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 2/8. Walter Unger 21.12.2018 12:38. Colouring of Interval-graphs (Algorithm) Theorem The graph-to-colour problem is for interval-graphs in time O(n log(n)) solvable. 1. Sort the intervals by their left endpoints.. 2. Check all endpoints e from the left to the right.. 3. If e is the starting point of an interval, colour it with the smallest free colour.. 4. If e is the ending point of an interval I is, free the colour of I .. Invariant If a node v is coloured with colour k, then v is part of a k-clique.. Z. Concluding Remarks SS2016.
(95) Intersection-Graphs 3:12. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 3/8. Walter Unger 21.12.2018 12:38. Colouring of Interval-graphs (Algorithm) Theorem The graph-to-colour problem is for interval-graphs in time O(n log(n)) solvable. 1. Sort the intervals by their left endpoints.. 2. Check all endpoints e from the left to the right.. 3. If e is the starting point of an interval, colour it with the smallest free colour.. 4. If e is the ending point of an interval I is, free the colour of I .. Invariant If a node v is coloured with colour k, then v is part of a k-clique.. Z. Concluding Remarks SS2016.
(96) Intersection-Graphs 3:12. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 4/8. Walter Unger 21.12.2018 12:38. Colouring of Interval-graphs (Algorithm) Theorem The graph-to-colour problem is for interval-graphs in time O(n log(n)) solvable. 1. Sort the intervals by their left endpoints.. 2. Check all endpoints e from the left to the right.. 3. If e is the starting point of an interval, colour it with the smallest free colour.. 4. If e is the ending point of an interval I is, free the colour of I .. Invariant If a node v is coloured with colour k, then v is part of a k-clique.. Z. Concluding Remarks SS2016.
(97) Intersection-Graphs 3:12. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 5/8. Walter Unger 21.12.2018 12:38. Colouring of Interval-graphs (Algorithm) Theorem The graph-to-colour problem is for interval-graphs in time O(n log(n)) solvable. 1. Sort the intervals by their left endpoints.. 2. Check all endpoints e from the left to the right.. 3. If e is the starting point of an interval, colour it with the smallest free colour.. 4. If e is the ending point of an interval I is, free the colour of I .. Invariant If a node v is coloured with colour k, then v is part of a k-clique.. Z. Concluding Remarks SS2016.
(98) Intersection-Graphs 3:12. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 6/8. Walter Unger 21.12.2018 12:38. Colouring of Interval-graphs (Algorithm) Theorem The graph-to-colour problem is for interval-graphs in time O(n log(n)) solvable. 1. Sort the intervals by their left endpoints.. 2. Check all endpoints e from the left to the right.. 3. If e is the starting point of an interval, colour it with the smallest free colour.. 4. If e is the ending point of an interval I is, free the colour of I .. Invariant If a node v is coloured with colour k, then v is part of a k-clique.. Z. Concluding Remarks SS2016.
(99) Intersection-Graphs 3:12. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 7/8. Walter Unger 21.12.2018 12:38. Colouring of Interval-graphs (Algorithm) Theorem The graph-to-colour problem is for interval-graphs in time O(n log(n)) solvable. 1. Sort the intervals by their left endpoints.. 2. Check all endpoints e from the left to the right.. 3. If e is the starting point of an interval, colour it with the smallest free colour.. 4. If e is the ending point of an interval I is, free the colour of I .. Invariant If a node v is coloured with colour k, then v is part of a k-clique.. Z. Concluding Remarks SS2016.
(100) Intersection-Graphs 3:12. Colouring. Interval-graphs. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 8/8. Walter Unger 21.12.2018 12:38. Colouring of Interval-graphs (Algorithm) Theorem The graph-to-colour problem is for interval-graphs in time O(n log(n)) solvable. 1. Sort the intervals by their left endpoints.. 2. Check all endpoints e from the left to the right.. 3. If e is the starting point of an interval, colour it with the smallest free colour.. 4. If e is the ending point of an interval I is, free the colour of I .. Invariant If a node v is coloured with colour k, then v is part of a k-clique.. Z. Concluding Remarks SS2016.
(101) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 1/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e. b a. d. g. k. f. l. c. i. j. m. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(102) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 2/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e. b a. d 1. g. k. f. l. c. i. j. m. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(103) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 3/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e. b a. d 1. g. k. f. l. c. i. j. m. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(104) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 4/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d. g. k. f. l. c. i. j. m. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(105) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 5/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d. g. k. f. l. c. i. j. m. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(106) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 6/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d c. g. k. f. l. 2. i. j. m. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(107) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 7/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d c. g. k. f. l. 2. i. j. m. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(108) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 8/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d c. 2. g. k. f. l. 2. i. j. m. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(109) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 9/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d c. 2. g. k. f. l. 2. i. j. m. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(110) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 10/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d c. 2 2. g. k. f. l. 2. i. j. m. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(111) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 11/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d c. 2 2. g. k. f. l. 2. i. j. m. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(112) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 12/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d c. g. 2 2. k 3. f 2. i. j. m. l n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(113) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 13/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d c. g. 2 2. k 3. f 2. i. j. m. l n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(114) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 14/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d c. g. 2 2. 3. 2. k. 3. f i. j. m. l n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(115) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 15/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d c. g. 2 2. 3. 2. k. 3. f i. j. m. l n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(116) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 16/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. m. l n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(117) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 17/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 h e 1. b a. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. m. l n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(118) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 18/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 3. h e 1. b a. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. m. l n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(119) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 19/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 3. h e 1. b a. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. m. l n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(120) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 20/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 3. h e 1. b a. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. m. l 4. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(121) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 21/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 3. h e 1. b a. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. m. l 4. n. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(122) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 22/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 3. h e 1. b a. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. m. l 4. n. 5. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(123) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 23/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 3. h e 1. b a. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. m. l 4. n. 5. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(124) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 24/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 3. h e 1. b a. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. 5. m. l 4. n. 5. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(125) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 25/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 3. h e 1. b a. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. 5. m. l 4. n. 5. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(126) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 26/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 3. h e 1. b a. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. l 4. n. 5. m 5 5. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(127) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 27/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 3. h e 1. b a. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. l 4. n. 5. m 5 5. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(128) Intersection-Graphs 3:13. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. Circle-Grahs. 28/28. Example of independent set problem on intervall-graphs 0. 2. 4. 6. 8. SS2016. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 3. h e 1. b a. Z. Concluding Remarks. Walter Unger 21.12.2018 12:38. 1. d c. g. 2 2. 3. 2. k. 3. f i. 3. j. l 4. n. 5. m. 6 5 5. 1. Sort the intervalls by their starting ponts.. 2. Go through all starting points e from left to right.. 3. Store for each intervall I the size of a maximal independet set of intervals, which contain I as the rightmost interval..
(129) Intersection-Graphs 3:14. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. 1/8. Circle-Grahs Walter Unger 21.12.2018 12:38. SS2016. Independent Set Problem for Interval-graphs Theorem Finding a maximal independet set is sovable in time O(n log(n)) on interval-graphs. 1. Sweep through the start- and endpoints of intervals from left to right.. 2. Store for each endpoint e the size of a maximal independent set of intervals, which is placed to the left of e. While sweeping from left to right do:. 3. 1. 2. 3. If e is a starting point of interval (e, f ) and there is no endpoint to the left of e, then let S(f ) = 1. If e is a starting point of interval (e, f )t, then compute: largest endpoint e 0 to the left of e and let S(f ) = S(e 0 ) + 1. If e is an endpoint of interval (a, e), then compute: largest endpoint e 0 to the left of e and to the right of a. If that exists, then let S(e) = max(S(e 0 ), S(e)).. Z. Concluding Remarks.
(130) Intersection-Graphs 3:14. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. 2/8. Circle-Grahs Walter Unger 21.12.2018 12:38. SS2016. Independent Set Problem for Interval-graphs Theorem Finding a maximal independet set is sovable in time O(n log(n)) on interval-graphs. 1. Sweep through the start- and endpoints of intervals from left to right.. 2. Store for each endpoint e the size of a maximal independent set of intervals, which is placed to the left of e. While sweeping from left to right do:. 3. 1. 2. 3. If e is a starting point of interval (e, f ) and there is no endpoint to the left of e, then let S(f ) = 1. If e is a starting point of interval (e, f )t, then compute: largest endpoint e 0 to the left of e and let S(f ) = S(e 0 ) + 1. If e is an endpoint of interval (a, e), then compute: largest endpoint e 0 to the left of e and to the right of a. If that exists, then let S(e) = max(S(e 0 ), S(e)).. Z. Concluding Remarks.
(131) Intersection-Graphs 3:14. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. 3/8. Circle-Grahs Walter Unger 21.12.2018 12:38. SS2016. Independent Set Problem for Interval-graphs Theorem Finding a maximal independet set is sovable in time O(n log(n)) on interval-graphs. 1. Sweep through the start- and endpoints of intervals from left to right.. 2. Store for each endpoint e the size of a maximal independent set of intervals, which is placed to the left of e. While sweeping from left to right do:. 3. 1. 2. 3. If e is a starting point of interval (e, f ) and there is no endpoint to the left of e, then let S(f ) = 1. If e is a starting point of interval (e, f )t, then compute: largest endpoint e 0 to the left of e and let S(f ) = S(e 0 ) + 1. If e is an endpoint of interval (a, e), then compute: largest endpoint e 0 to the left of e and to the right of a. If that exists, then let S(e) = max(S(e 0 ), S(e)).. Z. Concluding Remarks.
(132) Intersection-Graphs 3:14. Interval-graphs. Independent Sets and Cliques. Perm.-Gr.. Arc-Graphs. 4/8. Circle-Grahs Walter Unger 21.12.2018 12:38. SS2016. Independent Set Problem for Interval-graphs Theorem Finding a maximal independet set is sovable in time O(n log(n)) on interval-graphs. 1. Sweep through the start- and endpoints of intervals from left to right.. 2. Store for each endpoint e the size of a maximal independent set of intervals, which is placed to the left of e. While sweeping from left to right do:. 3. 1. 2. 3. If e is a starting point of interval (e, f ) and there is no endpoint to the left of e, then let S(f ) = 1. If e is a starting point of interval (e, f )t, then compute: largest endpoint e 0 to the left of e and let S(f ) = S(e 0 ) + 1. If e is an endpoint of interval (a, e), then compute: largest endpoint e 0 to the left of e and to the right of a. If that exists, then let S(e) = max(S(e 0 ), S(e)).. Z. Concluding Remarks.
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Planar graphs: no Intervall-graphs: yes Arc-graphs: no Permutation-graphs: yes Outerplanar graphs: no Maximal outerplanar graphs: yes Maximal planar graphs: no following slide
1-Way Gossip on Cycles Idea Messages should traverse in both directions.. Activate each f n-th node on
Gossip on arbitrary Trees Proof III Consider the two-way mode: by a similar way we may prove: At time t only two neighbours nodes u and v know the total information.. We get in
