Algorithmic Graph Theory (SS2016) Chapter 1 Planar Graphs Walter Unger

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(1)Algorithmic Graph Theory (SS2016) Chapter 1 Planar Graphs. Walter Unger Lehrstuhl für Informatik 1. 12:26 , December 21, 2018.

(2) Basic Definitions 1. Introduction to planar Graphs. Separators. Inhaltsverzeichnis. Contents I. 1. Basic Definitions Graphs Special Graphs Connectivity of Graphs Statements. 2. Introduction to planar Graphs Definitions Theorems on planar Graphs Definitions on outer-planar graphs Theorems on outer-planar Graphs Theorems on SP-Graphs Homeomorph Graphs. Z. Applications Walter Unger 21.12.2018 12:26. 3. Separators Motivation Definition Examples Alternative Definition Introduction to planar Separators Overview Preparation Overview Planare-Graph-Separator Theorem. 4. Applications Independent Set on planar Graphs. SS2016. x.

(3) Basic Definitions 1:1. Introduction to planar Graphs. Separators. Graphs. Definition: Graph v2. v9. Definition (Undirected Graph) Let V (G ) = {v1 , ..., vn } be a non-empty set of nodes and E (G ) be a set or multiset of pairs from V (G ) (set of edges). The sets V (G ) and E (G ) define the graph G = (V (G ), E (G )). If G is uniquely determined, then we just write: V and E . Or in other words G = (V , E ). We always use as default writing: n = |V | and m = |E |.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:26. v7. SS2016. g.

(4) Basic Definitions 1:2. Introduction to planar Graphs. Separators. Graphs. Way of Speaking for Graphs v2. v9. Definition (Way of Speaking) Let G = (V (G ), E (G )) and e = (v , w ) ∈ E (G ). The nodes v , w are called connected (adjacent) by an edge e.. v4 v6 v1 v8. An edge e is called loop, if v = w holds.. v5. Two edges are called parallel, if they are the same. A graph without parallel edges is called simple.. v0 v3. As long as we do not state differently we will use in the following simple graph without loops.. Z. Applications Walter Unger 21.12.2018 12:26. v7. SS2016. g.

(5) Basic Definitions 1:3. Introduction to planar Graphs. Separators. Graphs. Degree of a Node Definition (Degree of a Node) Let v ∈ V (G ). With deg(v ) = |{e ∈ E (G ) e = (v , v 0 ), v 0 ∈ V (G ) \ {v }}| we denote the degree of a Node (degree) of v . v2. deg(v0 ) = 4.. v9. deg(v1 ) = 3.. v4. deg(v4 ) = 6. deg(v5 ) = 6.. v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:26. v7. SS2016. g.

(6) Basic Definitions 1:4. Introduction to planar Graphs. Separators. Special Graphs. Regular and Complete Definition (Regular) A graph G is called k-regular, iff for all v ∈ V (G ) we have: d(v ) = k.. e d. a b. e f. d. c. a b. e f. d. c. a. f. c b. Definition (Complete) A graph G is called complete, iff all pairs of nodes a, b from V holds: (a, b) ∈ E . Notation: Kn .. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. g.

(7) Basic Definitions 1:5. Introduction to planar Graphs. Special Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. Special Graphs. SS2016. Definition (Bipartite) A Graph G is called bipartite, iff V may be split in to disjoint set V 0 , V 00 , such that each edge connects only nodes from both partitions. Notation: G = (V 0 , V 00 , E ) Definition (Complete bipartite) A Graph G is called complete bipartite, iff V may be split in to disjoint set V 0 , V 00 , and E = {(a, b) a ∈ V 0 , b ∈ V 00 }. Notation: Kp,q with p = |V 0 | and q = |V 00 |. Star, iff Sn = K1,n−1 .. g.

(8) Basic Definitions 1:6. Introduction to planar Graphs. Special Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. Examples. a4. b4. a3. b3. a3. b3. a2. b2. a2. b2. a1. b1. a1. b1. a0. b0. a0. b0. b4. SS2016. g.

(9) Basic Definitions 1:7. Introduction to planar Graphs. Separators. Special Graphs. Subgraphs v2. v9. Definition (Subgraph) A Graph H = (V (H), E (H)) is call a subgraph of G = (V (G ), E (G )),. v4 v6. iff V (H) ⊆ V (G ) and E (H) ⊆ E (G ). v1. v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:26. v7. SS2016. g.

(10) Basic Definitions 1:8. Introduction to planar Graphs. Separators. Special Graphs. Subgraphs v2. v9. Definition (node-induced subgraph) A graph H = (V (H), E (H)) is a node-induced subgraph of G = (V (G ), E (G )), iff V (H) ⊆ V (G ) and E (H) = {(a, b) ∈ E (G ) a, b ∈ V (H)}.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:26. v7. SS2016. g.

(11) Basic Definitions 1:9. Introduction to planar Graphs. Separators. Connectivity of Graphs. SS2016. Connectivity Definition A graph G = (V , E ) is called connected, iff between any two different nodes a, b exists a path from a to b. v2. v9 v4 v6. v1 v8 v5 v0 v3. v7. Z. Applications Walter Unger 21.12.2018 12:26. g.

(12) Basic Definitions 1:10. Introduction to planar Graphs. Connectivity of Graphs. Separators. Node-Separator Definition Let G = (V , E ), V 0 ⊂ V is called a node-separator (vertex cut), iff G − V 0 is not connected. Notation: G − V 0 := (V \ V 0 , {(a, b) ∈ E | a, b ∈ V \ V 0 }) Definition If {v } is a node-separator, then v is called articulation point. Theorem Only cliques Kn do not have any node-separator.. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. g.

(13) Basic Definitions 1:11. Introduction to planar Graphs. Separators Walter Unger 21.12.2018 12:26. Example. v2. v9. v2. v9 v4. v4 v6. v6. v1. v1 v8. v8. v5. v5. v0. v0 v3. v7. v2. v3. v7. v2. v9. v9 v4. v4. v6. v6 v1. v1. v8. v8 v5. v5 v0. v0 v3. v7. Z. Applications. Connectivity of Graphs. v3. v7. SS2016. g.

(14) Basic Definitions 1:12. Introduction to planar Graphs. Connectivity of Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. Edge-Separator Definition Let G = (V , E ). E 0 ⊂ E is called edge-separator (edge cut), iff G − E 0 is not connected. Notation: G − E 0 := (V , E \ E 0 ) Definition If {v , w } is an edge-separator, then {v , w } is called a bridge. Theorem An minimal edge-separator E 0 of G = (V , E ) induces a 2-partite graph. Or in other words: G = (V , E 0 ) is a 2-partite graph.. SS2016. g.

(15) Basic Definitions 1:13. Introduction to planar Graphs. Separators. Connectivity of Graphs. Example v2. v9. v4 v6 v1 v8 v5. v0 v3. v7. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. g.

(16) Basic Definitions 1:14. Connectivity of Graphs. Introduction to planar Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. Connectivity Definition A Graph G = (V , E ) is called k-connected, iff ∀V 0 ⊂ V : |V 0 | = k − 1 we have G − V 0 is connected. A k-connected Graph is also k − 1-connected. Notation: κ(G ) = k Definition. Let G = (V , E ) and k minimal with: ∃E 0 ⊂ E : |E 0 | = k and G − E 0 is not connected or trivial. Then we call G k-edge-connected. A k-edge-connected Graph is also k − 1-edge-connected. Notation: λ(G ) = k. g.

(17) Basic Definitions 1:15. Introduction to planar Graphs. Separators. Statements. Statements on Connectivity Theorem For any graph G = (V , E ) we have: κ(G ) 6 λ(G ) 6 δ(G ) Notation: δ(G ) := min{deg(v ) | v ∈ V } Theorem For all integer numbers 0 < a 6 b 6 c there are graphs G with: κ(G ) = a, λ(G ) = b, δ(G ) = c Theorem Let G = (V , E ) be a graph with: |V | = n and δ(G ) > n/2. Then we have: λ(G ) = δ(G ). Z. Applications Walter Unger 21.12.2018 12:26. SS2016. g.

(18) Basic Definitions 1:16. Introduction to planar Graphs. Separators. Statements. SS2016. Statements on Node-Connectivity Theorem Let G = (V , E ) with: |V | = n and |E | = m. Then is the maximal connectivity (maximal k with G is k-connected) of G : 0 falls m < n − 1 2 · m/n if m >n−1 Theorem Let G = (V , E ) connected. The following statements are equivalent: 1. v ∈ V is a node-separator.. 2. ∃a, b ∈ V : a, b 6= v : each path from a to b traverses via v . ˙ = V \ {v } and each path from a ∈ A to b ∈ B traverses via v . ∃A, B: A∪B. 3. Z. Applications Walter Unger 21.12.2018 12:26. g.

(19) Basic Definitions 1:17. Introduction to planar Graphs. Separators. Statements. Statements on Edge-Connectivity Theorem Let G = (V , E ) be connected. The following statements are equivalent: 1. e ∈ E is a edge-separator.. 2. e is not in any simple cycle of G .. 3. ∃a, b ∈ E : each path from a to b traverses via e. ˙ = V and each path from a ∈ A to b ∈ B traverses via e. ∃A, B: A∪B. 4. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. g.

(20) Basic Definitions 1:18. Introduction to planar Graphs. Separators. Statements. SS2016. Definition Definition Let G = (V , E ) and (a, b) = e ∈ E . The subdivision of an edge e results in graph ˙ }, E ∪ {(a, v ), (v , b)} \ {e}) G = (V ∪{v b. c. d. a. f. e. Z. Applications Walter Unger 21.12.2018 12:26. Definition A set of paths of G = (V , E ) is called intern-node-disjoint, iff no two paths share an internal-node. The internal nodes are all except the start and the end node.. g.

(21) Basic Definitions 1:19. Introduction to planar Graphs. Separators. Statements. Z. Applications Walter Unger 21.12.2018 12:26. Theorem Let G = (V , E ) with |V | > 3. The following statements are equivalent: 1. G is 2-connected.. 2. Each node pair is connected by two intern-node-disjoint paths.. 3. Each node pair is on a common simple cycle.. 4. There exits an edge and each node together with this edge is on a common simple cycle.. 5. There exit two edges and each pair of edges is on a common simple cycle.. 6. For each pair of nodes a, b and an edge e exists a simple path from a to b traversing e.. 7. For three nodes a, b, c exists a path from a to b traversing c.. 8. For three nodes a, b, c exists a path from a to b avoiding c.. SS2016. n.

(22) Basic Definitions 1:20. Introduction to planar Graphs. Statements. Separators. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. Statements Theorem Let G = (V , E ) k-connected. Then any k nodes are on a common simple cycle. Notation: Let (G = V , E ) and (H = W , F ) graphs ˙ , E ∪ F ∪ {(a, b) | a ∈ V , b ∈ W }) G + W = (V ∪W Theorem A graph G is 3-connected, iff G may be constructed from the weel Wi = K1 + Ci (i > 4) by the following operations: 1. Adding a new edge.. 2. Splitting a node of degree > 4 into two connected nodes of degree > 3.. g.

(23) Basic Definitions 1:21. Introduction to planar Graphs. Statements. Separators. Z. Applications Walter Unger 21.12.2018 12:26. Statements on k-Connectivity Theorem (Menger’s Theorem) G is k-connected, iff any two node are connected by k intern-node-disjoint paths. Theorem (Menger’s Theorem) G is k-edge-connected, iff any two node are connected by k edge-disjoint paths.. SS2016. g.

(24) Basic Definitions 1:22. Introduction to planar Graphs. Separators. Statements. Z. Applications Walter Unger 21.12.2018 12:26. Computing the Connectivity Theorem The 1-connectivity of a graph may be computed by DFS/BFS. Theorem The 1-edge-connectivity of a graph may be computed by DFS/BFS. Theorem The 2-connectivity of a graph may be computed by DFS/BFS. Theorem The k-connectivity of a graph may be computed by flow algorithms. Theorem The k-edge-connectivity of a graph may be computed by flow algorithms.. SS2016. g.

(25) Basic Definitions 1:23. Definitions. Introduction to planar Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. Definitions Definition A graph G = (V , E ) is called planar, iff it could be drawn in the plane without crossing edges. A connected area of such an embedding is called window. The unlimited window is called outer window. Definition A graph G = (V , E ) is called maximal planar, iff the adding of an edge makes G non-planar.. SS2016. g.

(26) Basic Definitions 1:24. Introduction to planar Graphs. Separators. Definitions. Example: planar Graph v0. v1. v5. v3. v6. v4. v2. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. g.

(27) Basic Definitions 1:25. Introduction to planar Graphs. Theorems on planar Graphs. Separators. SS2016. Results I Theorem If G = (V , E ) is planar and 2-connected, then each window is a simple cycle and each edge separates two different windows. Theorem (Euler) Let G = (V , E ) be a planar graph with |V | = n, |E | = m. Let f be the number of windows and k be the number of connected components. Then the following holds: n − m + f = 1 + k. Proof by simple induction.. Z. Applications Walter Unger 21.12.2018 12:26. i.

(28) Basic Definitions 1:26. Introduction to planar Graphs. Separators. Theorems on planar Graphs. Proof c2. n − m + f = 1 + k holds for a single node.. o2. new node: (n + 1) − m + f = 1 + (k + 1) new edge connects components: n − (m + 1) + f = 1 + (k − 1) or new edge seperates window: n − (m + 1) + (f + 1) = 1 + k.. a1. Z. Applications Walter Unger 21.12.2018 12:26. r1. e2. o1 c1. e1. SS2016. i.

(29) Basic Definitions 1:27. Theorems on planar Graphs. Introduction to planar Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. Results II. i. n−m+f = 1+k. Theorem Let G = (V , E ) be a planar graph with |V | = n, |E | = m and each window is a simple cycle of length k. Then the following holds: n−2 m=k· k −2 Note: k · f = 2 · m und n − m + f = 2 Theorem Let G = (V , E ) be a planar graph with |V | = n, |E | = m and each window is a 3-clique. Then the following holds: m = 3 · n − 6. If each window is a simple cycle of length 4, then we get: m = 2 · n − 4. Theorem Let G = (V , E ) be a planar graph with |V | = n > 3, |E | = m. Then we get: m 6 3 · n − 6. If G contains no triangles, then we have: m 6 2 · n − 4..

(30) Basic Definitions 1:28. Introduction to planar Graphs. Theorems on planar Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. Results III. SS2016. i. n−m+f = 1+k e 6 3·n−6. Theorem K5 and K3,3 are non-planar graphs. Theorem Let G = (V , E ) be a planar graph with |V | > 4. Then G contains at least four nodes with degree 6 5. Theorem Let G = (V , E ) be a planar graph. Then each window could become the outer window..

(31) Basic Definitions 1:29. Introduction to planar Graphs. Separators. Theorems on planar Graphs. Z. Applications Walter Unger 21.12.2018 12:26. Results IV. SS2016. i. n−m+f = 1+k e 6 3·n−6. Theorem Let G = (V , E ) be a maximal planar graph with |V | > 4. Then is G 3-conneceted. Theorem Each 3-connected planar graph is embeddable in a unique way on the spahre. Theorem Any planar graph could be drawn with straight lines on the plane..

(32) Basic Definitions 1:30. Introduction to planar Graphs. Separators. Theorems on planar Graphs. Recognition-Problem. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. i. n−m+f = 1+k e 6 3·n−6. Definition The following problem is the recognition-problem on graphs: Given a graph G = (V , E ) and a graph-class G. Question; does G ∈ G hold. Theorem The recognition-problem for planar graphs is sovable in linear time..

(33) Basic Definitions 1:31. Definitions on outer-planar graphs. Introduction to planar Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. Definition Definition A planar graph G is called outer-planar, iff it could be drawn without crossing in the plane, such that all nodes are on one (the outer) window. Definition A graph G = (V , E ) is called maximal outer-planar, iff the addition of any edge makes G non-outer-planar.. g.

(34) Basic Definitions 1:32. Introduction to planar Graphs. Separators. Definitions on outer-planar graphs. Example: outer-planar Graph v0. v1. v3. v2. v5. v4. v7 v6. v8. v9. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. g.

(35) Basic Definitions 1:33. Introduction to planar Graphs. Definitions on outer-planar graphs. Separators. SS2016. Definition Definition A planar graph G is called outer-planar, iff it could be drawn without crossing in the plane, such that all nodes are on one (the outer) window. Definition A graph G = (V , E ) is called maximal outer-planar, iff the addition of any edge makes G non-outer-planar. Definition A planar graph G = (V , E ) is called k-outer-planar, iff it could be drawn in the plane, such that no two edges cross and after deletion k − 1 times the nodes of the outer window, the remaining is a embedded outer-planar graph.. Z. Applications Walter Unger 21.12.2018 12:26. g.

(36) Basic Definitions 1:34. Introduction to planar Graphs. Theorems on outer-planar Graphs. Separators. SS2016. Results I Theorem Let G = (V , E ) be a maximal outer-planar graph with |V | = n > 3. Then G will have n − 2 inner windows. Theorem Let G = (V , E ) be a maximal outer-planar Graph with |V | = n and |E | = m. Then the following holds: 1. 2·n−3=m. 2. At least three nodes have a degree of 6 3.. 3. At least two nodes have a degree of two.. 4. G is exactly two-connected.. Theorem K4 and K2,3 are not outer-planar graphs.. Z. Applications Walter Unger 21.12.2018 12:26. i.

(37) Basic Definitions 1:35. Introduction to planar Graphs. Theorems on SP-Graphs. Separators. SS2016. SP-Graphs Definition A SP-graph is constructed by a sequence of series and parallel operations from the graphs ({a, b}, {(a, b)}) and ({a, b}, ∅). The parallel operation merges the corresponding connector nodes. The series operation merges two connector nodes. This new may not be used as a connector node in any future operation. Theorem K4 is not a SP-graph, but the K2,3 is a SP-graph.. Z. Applications Walter Unger 21.12.2018 12:26. g.

(38) Basic Definitions 1:36. Homeomorph Graphs. Introduction to planar Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. Definition Definition Two graphs G and H are called homeomorph, iff they could be constructed from the same graph by a sequence of subdivisions.. g.

(39) Basic Definitions 1:37. Introduction to planar Graphs. Homeomorph Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. Results I. SS2016. Theorem G = (V , E ) is outer-planar, iff no subgraph is homeomorph to the K4 or the K2,3 with the exception of the K4 − e. Theorem G = (V , E ) is a SP-graph, iff no subgraph is homeomorph to the K4 with the exception of the K4 − e. Theorem (Kuratowski) G = (V , E ) is planar, iff no subgraph is homeomorph to the K5 od K3,3 . Theorem A outer-planar graph is a SP-graph. A SP-Graph is a planar graph.. g.

(40) Basic Definitions 1:38. Introduction to planar Graphs. Homeomorph Graphs. Results I Theorem Any planar graph is 5-colourable. Theorem Any planar graph is 4-colourable. Theorem Any planar graph with at most two triangles is 3-colourable.. Separators. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. g.

(41) Basic Definitions 1:39. Introduction to planar Graphs. Homeomorph Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. A Proof Theorem Any planar graph is 5-colourable. Idea of Proof: Choose a node v of degree less than 6. Colour recursively G − {v }. If deg(v ) < 5 holds, v can be coloured. If all neighbours of v use just four colours, v can be coloured. If deg(v ) = 5 holds and all neighbours of v are coloured with different colours, note: Within G − {v } there is a component, which uses just two colours and can be recoloured. A short case discussion shows: There exists two colours and a component using these colours, such that just one neighbour of v receives a new colour.. SS2016. s.

(42) Basic Definitions 1:40. Introduction to planar Graphs. Separators. Homeomorph Graphs. Recolouring one Component a. b. di. bi. v e2. c2. e1. c1. e. c. d. b1. d2. d1. Z. Applications Walter Unger 21.12.2018 12:26. b2. SS2016. s.

(43) Basic Definitions 1:41. Homeomorph Graphs. Introduction to planar Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. Results II Theorem A planar graph is 4-colourable, iff each hamilton planar graph is 4-colourable. Theorem A planar graph is 4-colourable, iff each cubic planar graph without bridges is 3-colourable. Theorem The 3-colouring-problem on planar graphs if degree 6 4 is NP-complete.. SS2016. g.

(44) Basic Definitions 1:42. Introduction to planar Graphs. Separators. Homeomorph Graphs. Idea and Structure of Proof Theorem The 3-colouring-problem on planar graphs if degree 6 4 is NP-complete. Problem L1 is easyer then L2 : L1 6P L2 . If L2 is in P, then is also L1 in P. If L1 is hard, i.e. L1 ∈ N PC, then is also L2 ∈ N P. Structure of proof: Let L1 ∈ N PC and we assume L2 ∈ P. We transform input of L1 with function f into input for L2 such that: x ∈ L1 ⇐⇒ f (x) ∈ L2 . If f ∈ P holds, then we get L1 ∈ P, which is a contratiction.. Here we have: L1 is the 3-colouring-problem and L2 3-colouring-problem on planar graphs of degree 6 4.. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. s.

(45) Basic Definitions 1:43. Introduction to planar Graphs. Separators. Homeomorph Graphs. Idea and Structure of Proof Theorem The 3-colouring-problem on planar graphs if degree 6 4 is NP-complete. Let G = (V , E ) be the input of the 3-colouring-problem Construct planar f (G ) as input of the 3-colouring-problem Draw G in the plane. We get some crossings. Replace ech crossing with a 3-colorable planar graph, such that G is 3-colorable, f (G ) is 3-colorable.. Lemma There exists a planar graph H with nodes a, c, b, d: The nodes a, c, b, d are on the outer face in that order. The nodes a, b take in any 3-coloring of H the same color. The nodes c, d take in any 3-coloring of H the same color.. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. s.

(46) Basic Definitions 1:44. Introduction to planar Graphs. Separators Walter Unger 21.12.2018 12:26. Proof (planar) The central nodes are coloured w.l.o.g. as follows.. b. Case 1: Colour a blue.. d. l. h. k. i. e. g. m. f. j. a. Z. Applications. Homeomorph Graphs. c. SS2016. s.

(47) Basic Definitions 1:45. Introduction to planar Graphs. Separators Walter Unger 21.12.2018 12:26. Proof (planar 2.case) The central nodes are coloured w.l.o.g. as follows.. b. Case 2: Colour a red.. d. l. h. k. i. e. g. m. f. j. a. Z. Applications. Homeomorph Graphs. c. SS2016. s.

(48) Basic Definitions 1:46. Introduction to planar Graphs. Separators. Homeomorph Graphs. Proof (planar) Each crossing is replaced by such a component.. x1 d1. y2. z2. b1. b2. l1. h1. k1. i1. e1. g1. m1. f1. j1. ya1 1. c1 d2. l2. h2. k2. i2. e2. g2. m2. f2. j2. z1 a2. c2. Z. Applications Walter Unger 21.12.2018 12:26. x2. SS2016. s.

(49) Basic Definitions 1:47. Introduction to planar Graphs. Separators. Homeomorph Graphs. Proof (planar, degree 4) There exists a component H with three nodes a, h, d of degree 2 which are coloured the same in each 3-colouring of H.. h. e. a. g. f. b. Z. Applications Walter Unger 21.12.2018 12:26. c. d. SS2016. s.

(50) Basic Definitions 1:48. Introduction to planar Graphs. Separators. Homeomorph Graphs. Proof (planar, degree 4) There exists a component Hx with x nodes of degree 2 which are coloured the same in each 3-colouring of Hx . h1. a1. h2. g1. f1. e1. b1. c1. a2 d1. g2. f2. e2. b2. Z. Applications Walter Unger 21.12.2018 12:26. c2. d2. SS2016. s.

(51) Basic Definitions 1:49. Introduction to planar Graphs. Separators. Homeomorph Graphs. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. Summary (Proof) Replace edge-crossings by the above construction, such that each crossing is replaced by one component. I.e. an edge with x crossings will be replaced by x components and one edge.. Replace a node of degree g > 4 by d(g − 6)/2e + 1 components of the second construction. Note: x tree-wise connected components have x + 2 nodes of degree 2 coloured by the same colour. 2 · (d(g − 6)/2e + 1 + 2) > 2 · ((g − 6)/2 + 3) = g. s.

(52) Basic Definitions 1:50. Introduction to planar Graphs. Separators. Motivation. Z. Applications Walter Unger 21.12.2018 12:26. Introduction Basis for all divide and conquer algorithmns.. 06. 16. 26. 36. 46. 56. 66. We would like to have small separators.. 05. 15. 25. 35. 45. 55. 65. Split the graph at the separator.. 04. 14. 24. 34. 44. 54. 64. Solve the problem recursively on the disconnected components. 03. 13. 23. 33. 43. 53. 63. Construct the solution by using the sub-solutions.. 02. 12. 22. 32. 42. 52. 62. 01. 11. 21. 31. 41. 51. 61. 00. 10. 20. 30. 40. 50. 60. Here: separators for planar graphs.. SS2016. g.

(53) Basic Definitions 1:51. Introduction to planar Graphs. Definition. Separators. SS2016. Definition Definition Let G = (V , E ) be a graph and n = |V |.. Z. Applications Walter Unger 21.12.2018 12:26. 6 α·n. 6 α·n. Let 0 6 α 6 1 be a constant. Let f (n) be a function. We call C ⊂ V a (f (n), α)-separator, iff. 6 f (n). |C | 6 f (n) and each component of G [V \ C ] contains at most α · n nodes. 6 α·n. 6 α·n. g.

(54) Basic Definitions 1:52. Introduction to planar Graphs. Examples. Separators. Example 1 Lemma A tree T has a (1, 1/2)-separator. Proof: Choose a arbitrary node c as a candidate. Let Ti be the trees in T − c. If one component Ti contains more than n/2 nodes, then choose a new candidate c := Γ(c) ∩ V (Ti ). After such a step the size of the largest component decreases by at least one. Repeat till a separator is found.. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. g.

(55) Basic Definitions 1:53. Introduction to planar Graphs. Separators. Examples. Example Outer-planar Graph a4. c6. a3. e4. c5. s5. e3. c4. a2. a1. u4. s4. u3. s2. u2. s3 r3. c3. c2. e2. c1. e1. r2 r1. s1. Z. Applications Walter Unger 21.12.2018 12:26. u1. SS2016. g.

(56) Basic Definitions 1:54. Introduction to planar Graphs. Examples. Example 2 Lemma A outer-planar graph G has a (3, 1/2)-separator. Proof: Maximise the outer-planar graph G . Use the above technique. Thus use the tree of inner windows. Choose as separator the node of the selected window.. Separators. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. g.

(57) Basic Definitions 1:55. Introduction to planar Graphs. Separators. Examples. Example Tree 03. 13. 23. a4. c6. a3 02. 33. 43. e4. c5 22. 42. 11. 21. 31. 51 s2. r1. e1 20. 41 r2. e2. c1 10. 62. r3. c2. a1 00. u3 52. s3. c3. a2. 63 u4. s4 32. c4. 01. 53 s5. e3. 12. 30. 61 u2. s1 40. Z. Applications Walter Unger 21.12.2018 12:26. u1 50. 60. SS2016. g.

(58) Basic Definitions 1:56. Introduction to planar Graphs. Alternative Definition. Separators. Alternative Definition Definition Let G = (V , E ) be a graph and n = |V |. Let f (n) be a function. Then C ⊂ V is called a f (n)-separator, iff. Z. Applications Walter Unger 21.12.2018 12:26. 6 2/3 · n. V may be split in C , T1 , T2 . |C | 6 f (n). T1 , T2 are not connected. Ti has at most 2/3 · n nodes. 6 f (n). 6 2/3 · n. SS2016. g.

(59) Basic Definitions 1:57. Introduction to planar Graphs. Separators. Alternative Definition. Z. Applications Walter Unger 21.12.2018 12:26. Comparing above Definitions Lemma G has a (f (n), 2/3)-separator, iff G has a f (n)-separator. Show ⇐= Each component K contains at most in one Ti . Thus |V (Ti )| 6 2/3 · n holds. Show =⇒ If a component K contains at least 1/3 · n nodes. then choose T1 = K . If all components contain less then 1/3 · n nodes, then enlarge T1 step by step till T1 contains more than 1/3 · n nodes. Then T2 contains at most 2/3 · n nodes.. SS2016. s.

(60) Basic Definitions 1:58. Introduction to planar Graphs. Separators. Z. Applications. Introduction to planar Separators. Walter Unger 21.12.2018 12:26. Introduction Planar graphs are important with many applications. How large could be a minimal separator in a planar Graph? First example: 09. 19. 29. 39. 49. 59. 69. 79. 89. 99. 08. 18. 28. 38. 48. 58. 68. 78. 88. 98. 07. 17. 27. 37. 47. 57. 67. 77. 87. 97. 06. 16. 26. 36. 46. 56. 66. 76. 86. 96. 05. 15. 25. 35. 45. 55. 65. 75. 85. 95. 04. 14. 24. 34. 44. 54. 64. 74. 84. 94. 03. 13. 23. 33. 43. 53. 63. 73. 83. 93. 02. 12. 22. 32. 42. 52. 62. 72. 82. 92. 01. 11. 21. 31. 41. 51. 61. 71. 81. 91. 00. 10. 20. 30. 40. 50. 60. 70. 80. 90. SS2016. s.

(61) Basic Definitions 1:59. Introduction to planar Graphs. Introduction to planar Separators. Introduction Planar graphs are important with many applications. There is no separator of constant size. √ Aim: O( n)-separator. Consider maximal planar graphs. Consider cycles as separators.. Separators. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. s.

(62) Basic Definitions 1:60. Introduction to planar Graphs. Overview. Separators. Overview Theorem (Lipton, Tarjan 1979) Each panar graph with n nodes has a (2 ·. √ 2n, 2/3)-separator.. Theorem (Lipton, Tarjan 1979) √ A (2 · 2n, 2/3)-separator can be constructed on planar graphs in time O(n). Theorem (Lipton, Tarjan 1979) Let G = (V , E ) be a planar ε 6p1 with ε · n > 1. p graph and √ Then contains G a ((2 + 2/(ε · n)) · 6 n/ε, ε)-separator, which could be constructed in time O(n log 1/ε).. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. s.

(63) Basic Definitions 1:61. Introduction to planar Graphs. Separators. Preparation. Basic Idea 2. We could hope for a good separator. But in general we may √ need O( n). In the worse case the planar graph is maximal.. 1. Z. Applications Walter Unger 21.12.2018 12:26. 17. 27. 37. 47. 57. 67. 77. 16. 26. 36. 46. 56. 66. 76. 15. 25. 35. 45. 55. 65. 75. 14. 24. 34. 44. 54. 64. 74. 13. 23. 33. 43. 53. 63. 73. 12. 22. 32. 42. 52. 62. 72. 11. 21. 31. 41. 51. 61. 71. SS2016. w.

(64) Basic Definitions 1:62. Introduction to planar Graphs. Separators. Preparation. Preparation Definition (Diameter and Radius) The diameter of G = (V , E ) is: diam(G ) = max{dist(v , w ) | v , w ∈ V }. The radius of a node v ∈ V is: rad(v , G ) = max{dist(v , x) | x ∈ V } The radius of G is: rad(G ) = min{rad(v , G ) | v ∈ V }.. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. w.

(65) Basic Definitions 1:63. Introduction to planar Graphs. Separators. Preparation. Preparation Lemma Let G = (V , E ) be a planar graph and B = (V , T ) be a spanning-tree of G with radius s. Then G contains a (2 · s + 1, 2/3)-separator. Proof: Let G be triangulated and embedded in the plane as a planar Graph. Let e ∈ E \ T . e assembles with some edges from T a unique cycle Ce . By int(Ce ) we denote the number of nodes which are inside Ce . ext(Ce ) we denote the number of nodes which are outside Ce . Aim: Search e with int(Ce ) 6 2/3 · n and ext(Ce ) 6 2/3 · n. Then is Ce a (2 · s + 1, 2/3)-separator.. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. w.

(66) Basic Definitions 1:64. Introduction to planar Graphs. Preparation. 2 17. 27. 37. 47. 57. 67. 77. 16. 26. 36. 46. 56. 66. 76. 15. 25. 35. 45. 55. 65. 75. 14. 24. 34. 44. 54. 64. 74. 13. 23. 33. 43. 53. 63. 73. 12. 22. 32. 42. 52. 62. 72. 11. 21. 31. 41. 51. 61. 71. Z. Applications Walter Unger 21.12.2018 12:26. Example. 1. Separators. SS2016. w.

(67) Basic Definitions 1:65. Introduction to planar Graphs. Separators. Preparation. SS2016. Proof (continued) Search step by step for an edge e with int(Ce ) 6 2/3 · n and ext(Ce ) 6 2/3 · n. Choose any e. If int(Ce ) 6 2/3 · n and ext(Ce ) 6 2/3 · n holds, terminate. Let w.lo.g.: int(Ce ) > 2/3 · n. Let e = {x, y } and z be the missing node of the window attached at e and in the inside of Ce . If e 0 = {x, z} on the cycle Ce , continue with considering Ce 00 . If e 00 = {y , z} on the cycle Ce , continue with considering Ce 0 . Otherwise let w.l.o.G. int(Ce 0 ) 6 int(Ce 00 ) and consider now Ce 00 .. In the last step int(Ce ) 6 2/3 · n und int(Ce ) > 1/3 · n holds. It follows that int(Ce ) 6 2/3 · n und ext(Ce ) 6 2/3 · n holds.. Z. Applications Walter Unger 21.12.2018 12:26. w.

(68) Basic Definitions 1:66. Introduction to planar Graphs. Preparation. Separators. Proof (continued) Last step in detail: The inside of e = {x, y } is too large: 2/3 · n. < =. int(C{x,y } ) int(C{x,z} ) + int(C{y ,z} ) + |C{x,z} ∩ C{y ,z} | − 1. The inside of e 00 = {x, z} is the larger part of (int(Ce 0 ) 6 int(Ce 00 )): 2/3 · n. < 6. int(C{x,z} ) + int(C{y ,z} ) + |C{x,z} ∩ C{y ,z} | − 1 2 · int(C{x,z} ) + |C{x,z} |. This way we get: ext(C{x,z} ). = < =. Z. Applications Walter Unger 21.12.2018 12:26. n − |C{x,z} | − int(C{x,z} ) n − 1/3 · n 2/3 · n. SS2016. w.

(69) Basic Definitions 1:67. Introduction to planar Graphs. Planare-Graph-Separator Theorem. Separators. √ 2 · l + 1 6 2 2n. Example. r. Z. Applications Walter Unger 21.12.2018 12:26. Start BFS from √ some node r . If the radius is smaller than 2n we apply the lemma.. SS2016. w.

(70) Basic Definitions 1:68. Introduction to planar Graphs. Planare-Graph-Separator Theorem. Separators. Z. Applications Walter Unger 21.12.2018 12:26. Example. r. √ Consider the case that the radius is largen then 2n. Each intermediate level disconnects the graph. We could only hope for a small separator.. SS2016. w.

(71) Basic Definitions 1:69. Introduction to planar Graphs. Planare-Graph-Separator Theorem. Separators. Z. Applications Walter Unger 21.12.2018 12:26. Example. r. None of the levels is a separator. p Check set of levels of distance s = d n/2e. One set is smaller then bn/sc.. SS2016. w.

(72) Basic Definitions 1:70. Introduction to planar Graphs. Planare-Graph-Separator Theorem. Separators. Z. Applications Walter Unger 21.12.2018 12:26. Example. r. If this set is no separator, conider the largest component. And apply the lemma.. SS2016. w.

(73) Basic Definitions 1:71. Introduction to planar Graphs. Planare-Graph-Separator Theorem. Separators. Planarer Separator (Teil 1) Theorem (Lipton, Tarjan 1979) Any planar Graph with n nodes has a (2 ·. Z. Applications Walter Unger 21.12.2018 12:26. √ 2n, 2/3)-separator.. Proof: Choose node w as the root. Determine Si (1 6 i 6 l) the set of nodes at distance i from w . √ If 2 · l + 1 6 2 2n holds, the proof follows from the above Lemma. p Otherwise let s = d n/2e. Define Lj = ∪i≡j mod s Si for 0 6 j < s. For a k hold: |Lk | 6 bn/sc. Consider H = G [V \ Lk ]. Assume now, that one component of H has more than 2/3 · n nodes.. SS2016. w.

(74) Basic Definitions 1:72. Introduction to planar Graphs. Planare-Graph-Separator Theorem. Separators. Proof (continued) H contains at most s − 1 continuous levels Si . Let Sl , Sl+1 , . . . , Sl+s−2 be those levels. Show that H could be embedded as a planar graph H 0 with radius s − 1. If l = 0 holds, is w part of H and we have an embedding. Otherwise l > 0 holds and we connect all nodes from Sl with a node w 0 .. We have by the above lemma for H 0 a (2 · s − 1, 2/3)-separator C 0 . The separator for G is C = C 0 ∪ Lk . |C | 6 bn/sc + 2 · s − 1. p Note: s = d n/2e. √ √ Thus we have: |C | 6 2n + 2n.. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. w.

(75) Basic Definitions 1:73. Introduction to planar Graphs. Planare-Graph-Separator Theorem. Separators. Theorem 2 Theorem (Lipton, Tarjan 1979) √ A (2 · 2n, 2/3)-separator for a planar graph may be computed in time O(n). Computing the levels: breath-first-search. Counting the nodes: run through a tree. Planar embedding: depth-first-search. Triangulation: local search. Construction of the tree of windows: depth-first-search. Counting the nodes: run through the tree of windows Used also: dynamic programming.. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. w.

(76) Basic Definitions 1:74. Introduction to planar Graphs. Planare-Graph-Separator Theorem. Separators. Z. Applications Walter Unger 21.12.2018 12:26. Resultats Theorem Any graph with √ genus g and n nodes has a √ (6 · gn + 2 · 2n + 1, 2/3)-separator, which could be computed in time O(n + g ). Theorem √ Any graph without H-minor and n nodes has a (|H|3/2 n, 2/3)-separator.. SS2016. g.

(77) Basic Definitions 1:75. Introduction to planar Graphs. Independent Set on planar Graphs. Separators. Z. Applications Walter Unger 21.12.2018 12:26. NPC Theorem The independent set problem on planar Graphs of degree three is NP-complete. Proof: Construct component for the crossing of two edges. This component will increase the size of the independent set by six. We could replace a polynomial number of crossing with this component. Replace a node of degree > 4 by a special binary tree. The leaves will take the role of the original node. There could be two cases: All leaves are in the independent set and the total number within the tree is x. No leave is in the independent set and the total number within the tree is x − 1.. SS2016. n.

(78) Basic Definitions 1:76. Introduction to planar Graphs. Separators. Independent Set on planar Graphs. Crossings (1) a. e. g. f. i. h. j c. k o. l m. n. p. r. q. s. b. t. Z. Applications Walter Unger 21.12.2018 12:26. d. SS2016. n.

(79) Basic Definitions 1:77. Introduction to planar Graphs. Separators. Independent Set on planar Graphs. Crossings (2a) a. e. g. f. i. h. j c. k o. l m. n. p. r. q. s. b. t. Z. Applications Walter Unger 21.12.2018 12:26. d. SS2016. n.

(80) Basic Definitions 1:78. Introduction to planar Graphs. Separators. Independent Set on planar Graphs. Crossings (2b) a. e. g. f. i. h. j c. k o. l. m. n p. r. q. s. b. t. Z. Applications Walter Unger 21.12.2018 12:26. d. SS2016. n.


(82) Basic Definitions 1:80. Introduction to planar Graphs. Separators. Independent Set on planar Graphs. Crossings (3b) a. e. g. f. i. h. j c. k o. l m. n. p. r. q. s. b. t. Z. Applications Walter Unger 21.12.2018 12:26. d. SS2016. n.


(84) Basic Definitions 1:82. Introduction to planar Graphs. Separators. Independent Set on planar Graphs. Crossings (4b) a. e. g. f. i. h. j c. k o. l m. n. p. r. q. s. b. t. Z. Applications Walter Unger 21.12.2018 12:26. d. SS2016. n.

(85) Basic Definitions 1:83. Introduction to planar Graphs. Separators. Independent Set on planar Graphs. Gradkomponente g0. f0. f1. e0. e1. Z. Applications Walter Unger 21.12.2018 12:26. d0. d1. d2. d3. c0. c1. c2. c3. b0. b1. b2. b3. b4. b5. b6. b7. a0. a1. a2. a3. a4. a5. a6. a7. SS2016. n.

(86) Basic Definitions 1:84. Introduction to planar Graphs. Separators Walter Unger 21.12.2018 12:26. Algorithm Theorem √. The independet set problem on planar graphs is solvable in time 2O(. n). Algorithm: Compute a C . For each independent set I on C : Remove all nodes Γ(I ) from the components of G [V \ C ]. Solve the independent set problem recursively on each component.. Running time: t(n) 6 O(n) + 2. √ 8n. + O(n) · t(2/3 · n).. Let k1 , k2 , n0 be the constants of the O terms, √. I.e. (k1 + k2 ) · n 6 2. 8n. for all n > n0 .. Let L > t(n) for all n 6 n0 .. √ 2 √8n 2/3. Show by induction: t(n) 6 L · 2 1−. .. Z. Applications. Independent Set on planar Graphs. .. SS2016. n.

(87) Basic Definitions 1:85. Introduction to planar Graphs. Separators. Independent Set on planar Graphs. Proof (Continuation) √ 2 √8n 2/3. Show by induction: t(n) 6 L · 2 1−. .. Holds for n 6 n0 . Let n > n0 :. √. Reminder: (k1 + k2 ) · n 6 2. 8n. . √. t(n). 6 6. k1 √ · n + 2 8n + k2 · n · t(2/3 · n) 2· 8n · t(2/3√· n) 2. 6. 22·. =. L·2. √. 2. 8n. 2/3·8n √ 2/3. · L · 2 1−. √ 2 √8n 1− 2/3. Z. Applications Walter Unger 21.12.2018 12:26. SS2016. n.

(88) Questions 2. Inhaltsverzeichnis. Walter Unger 21.12.2018 12:26. Legend n : Not of relevance g : implicitly used basics i : idea of proof or algorithm s : structure of proof or algorithm w : Full knowledge. SS2016. Z. x.

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