Algorithmic Graph Theory (SS2016)

Transcription

1 Algorithmic Graph Theory (SS2016) Chapter 4 Treewidth and more Walter Unger Lehrstuhl für Informatik 1 19:00, June 15, 2016

2 4 Inhaltsverzeichnis Walter Unger :00 SS2016 Z Contents I 1 Bandwidth Motivation Definition Problems Complete Trees 2 Pathwidth Idea Definition Theorems 3 Treewidth Definition Theorems Vertex Cover 4 k-trees Example Definition Theorems 5 Applications Cactus and near-trees Halin-graphs and outer-planar graphs SP-Graphs Minors

3 4:1 Motivation Walter Unger :00 SS2016 Z Motivation Till now: Problems are efficient solvable, if the flow of information is not too large. Example: interval-graphs, permutation-graphs, trees,... Idea: Try to generalize the restricted flow of information of the trees. Define a generalized tree. Idea for this: make the nodes fat. We start the bandwidth problem. After that: pathwidth, treewidth and partial k-trees.

4 4:2 Definition Walter Unger :00 SS2016 Z Definition of Bandwidth Definition (Bandwidth) Let G = (V, E) be a graph and let v, v V. A labeling of G is a function e : V N with e(v) = e(v ) v = v. The distance between v and v in the labeling e is given by: dist(e, v, v ) = e(v) e(v ). The bandwidth of the labeling e on G is bw(e, G) = max{dist(e, v, v ) {v, v } E}. The bandwidth of graph G is bw(g) = min e:v N {bw(e, G)}.

5 4:3 Definition Walter Unger :00 SS2016 Z Example e(v2) = 2 v2 e(v2) = 1 v2 e(v1) = 1 v1 v3 e(v3) = 3 e(v1) = 2 v1 v3 e(v3) = 3 e(v6) = 6 v6 Σ = 0 v5 e(v5) = 5 bandwidth = 5 v4 e(v4) = e(v6) = 4 v6 Σ = 0 v5 e(v5) = 6 bandwidth = 2 v4 e(v4) =

6 4:4 Definition Walter Unger :00 SS2016 Z Example: second view e(v2) = 2 v2 e(v2) = 1 v2 e(v1) = 1 v1 v3 e(v3) = 3 e(v1) = 2 v1 v3 e(v3) = 3 e(v6) = 6 v6 v4 e(v4) = 4 Σ = 0 v5 e(v5) = 5 bandwidth = 5 e(v6) = 4 v6 v4 e(v4) = 5 Σ = 0 v5 e(v5) = 6 bandwidth = 2 Σ = 0 v2 v1 v3 v6 v4 v5 Σ = 0 v1 v2 v3 v4 v5 v6

7 4:5 Problems Walter Unger :00 SS2016 Z Bandwidth Definition (Bandwidth-Problem) The bandwidth-problem for a graph is: Input: A graph G = (V, E) and a k N. Output: Does bw(g) k hold? Theorem The bandwidth-problem is NP-complete.

8 4:6 Problems Walter Unger :00 SS2016 Z Bandwidth on Caterpillars Definition (Caterpillar) A Caterpillar is a tree where all nodes of degree 3 are on a path. Σ = 0 v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 a1 a2 a3 a4 a5 a6 a7 a8 c3 c4 c5 c7 c8 e4 e5 Theorem The bandwidth-problem is NP-complete on caterpillars.

9 4:7 Problems Walter Unger :00 SS2016 Z k-bandwidth Definition (bandwidth-problem) The k-bandwidth-problem on a graph is: Input: A graph G = (V, E). Output: Does bw(g) k hold? Theorem The k-bandwidth-problem can be solved in linear time. Theorem Let G = (V, E) be a graphs with bw(g) = k, the following problem may be solved in linear time: Independent-Set, Clique, Vertex-Cover Colouring-problem Hamilton-Cycle, Hamilton-Path

10 4:8 Problems Walter Unger :00 SS2016 Z Idea for this Let bw(g) = k. Let the nodes be sorted by the labeling. I.e e(v i ) = i. Consider block B i = {v i, v i+1, v i+2,..., v i+k }. There is no edge from a node to the left of B i to a node on the right of B i I.e. there is no edge from a node v a to a node v b with a < i and b > i + k. This means: any information must pass B i. This calls for a solution using dynamic programing. Code on B i all possible solution for v 1, v 2,..., v i+k. Compute all possible solutions for the nodes v 1, v 2,..., v i+k+1 by using the data on B i and code them on B i+1.

11 4:9 Problems Walter Unger :00 SS2016 Z 3-Colouring Σ = 0 v0 v1 v2 v3 v4 v5 v6 v7 v8 v9

12 4:10 Problems Walter Unger :00 SS2016 Z g-colouring Input: G = (V, E) with bw(g) k: V = {v 1, v 2,..., n} and E {{v i, v j } i < j i + k} Data structure C i is definied by: (c 0, c 1,... c k ) C i g-colouring c of {v 1, v 2,..., v i+k } : j{0,..., k} : c j = c(v i+j ) Compute C 1 by: (c 0, c 1,... c k ) C 1 g-colouring c of {v 1, v 2,..., v 1+k } : j {0,..., k} : c j = c(v 1+j ) Compute C i+1 from C i by: (c 0, c 1,... c k ) C i+1 c : (c, c 0, c 1,..., c k 1 ) C i j {0,..., k 1} : {v i+j, v i+k } E c i c k

13 4:11 Problems Walter Unger :00 SS2016 Z Independent Set v0 v1 v2 v3 v4 v5 v6 v7 v8 v v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 Σ = 0

14 4:12 Problems Walter Unger :00 SS2016 Z Independent Set Input: G = (V, E) with bw(g) k: V = {v 1, v 2,..., n} and E {{v i, v j } i < j i + k} Data structure C i is defined by: (I, s) C i S {v 1,..., v i+k } : S {v i,... v i+k } = I, S = s, S is independent set Compute C 1 by: (I, s) C 1 I {v 1,..., v 1+k } : I = s, I is independent set Compute C i+1 from C i by: (I, s) C i+1 (I, s ) C i I = I \ {v i }, s = s or I = (I {v i+k+1 }) \ {v i }, s = s + 1, I is stable set

15 4:13 Problems Walter Unger :00 SS2016 Z Hamilton Cycle Σ = 0 v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 Open Endpoint Visited Node Nonvisited Node

16 4:14 Problems Walter Unger :00 SS2016 Z Hamilton-Cycle Input: G = (V, E) with bw(g) k: V = {v 1, v 2,..., n} and E {{v i, v j } i < j i + k} Data structure C i describes [0,2]-factors in {v 1, v 2,..., v i+k }: H = ({v 1, v 2,..., v i+k }, F ) is a [0,2]-factor in {v 1, v 2,..., v i+k }. δ H (v j ) = 2 for all j {1, 2,..., i 1}. I.e. each component in H is path. For each component C j, i j i + k : δ H (v j ) = 1 und v j C. I.e. each component in H is a path with endpoints in {v i, v i+1,..., v i+k }. Problem has solution, if H = ({v 1, v 2,..., v i+k }, F ) is [1,2]-Factor in {v 1, v 2,..., v n}. a, b : n k a, b n: j {1, 2,..., n} \ {a, b} : δ H (v a) = 2. δ H (v a) = δ H (v b ) = 1. {v a, v b } E.

17 4:15 Problems Walter Unger :00 SS2016 Z Lower Bound on Bandwidth 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 }.

18 4:16 Complete Trees Walter Unger :00 SS2016 Z Lower Bound for Bandwidth Theorem (Lower Bound for Bandwidth) Let G = (V, E) be a graph with n = V nodes. Then the following hold: n 1 bw(g) diam(g) Theorem (Lower Bound for Bandwidth of a Complete Tree) Let T = (V, E) be a complete tree with depth k. Then the following hold: 2 k 1 2 k+1 2 bw(g) =. k 2k

19 4:17 Complete Trees Walter Unger :00 SS2016 Z Bandwidth for Complete Trees Theorem (Upper Bound for Bandwidth of the Complete Binary Tree) Let T = (V, E) be a complete binary tree with depth k, then the following hold: 2 k 1 bw(t ) =. k

20 4:18 Complete Trees Walter Unger :00 SS2016 Z Hardness of the Bandwidth Problem Theorem For ε > 0 it is not possible to approximate the bandwidth-problem by a factor of 2 ε, under the assumption P N P. Theorem It is not possible to approximate the bandwidth-problem by a constant factor of k, under the assumption P N P. Theorem It is not possible to approximate the bandwidth-problem for caterpillars by a constant factor of k, under the assumption P N P.

21 4:19 Complete Trees Walter Unger :00 SS2016 Z Hardness of Bandwith (Idea) a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22

22 4:20 Complete Trees Walter Unger :00 SS2016 Z Hardness of Bandwith (Idea) a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22

23 4:21 Complete Trees Walter Unger :00 SS2016 Z Hardness of Bandwith (Idea) 14a15a16a17a18 c2 a19a20a21a22a23a24a25a26a27 c3 a28a29a30a31a32a33a34a35a36 c4 a37a38a39a40a4

24 4:22 Complete Trees Walter Unger :00 SS2016 Z Hardness of Bandwith (Idea)

25 4:23 Idea Walter Unger :00 SS2016 Z Idea of Pathwidth 00 Σ =

26 4:24 Definition Walter Unger :00 SS2016 Z Pathwidth Definition A graph G = (V, E) has pathwidth k, iff there is a path P = (V p, E p) and a mapping f : V p P(V ) with: (a, b) E : x V p : a, b f (x) If c is on the path from a to b on P, then does f (b) f (a) f (c) hold. x V p : f (x) k + 1. and for k 1 exists no such function f and path P. Notation: pw(g) = k. Theorem Let G = (V, E) be a graph. Then holds: bw(g) pw(g). Note: bw(k 1,2n+1) = n but pw(k 1,2n+1) = 1

27 4:25 Theorems Walter Unger :00 SS2016 Z Theorems I Theorem Let G = (V, E) be a graph. Then does bw(g) pw(g) holds. Let G = (V, E) be a graph with bw(g) = k and V = n. Show pw(g) k. Let e be an embedding function with bw(e, G) = k. Let e(v i ) = i. Let P n k = ({p 1, p 2,..., p n k+1 }, {{p j, p j+1 } 1 j n k}) be a path with n k + 1 nodes. Define function f : {p 1, p 2,..., p n k+1 } P(V ): f (p i ) = {v i, v i+1,..., v i+k }. Then the following holds: f (p i ) = k + 1. And if {v i, v i+d } E hold, then {v i, v i+d } f (p i ) follows. Thus we have: pw(g) k

28 4:26 Theorems Walter Unger :00 SS2016 Z Theorems II Theorem Let G = (V, E) be a graph. Then the following hold: The problem, to compute the pathwidth of a graph, is NP-complete. For a fixed k it is possible to check in linear time O(n + m), if a graph has pathwidth k. Theorem Let G = (V, E) be a graphs with pw(g) = k. The following problem may be solved in linear time: Independent-Set, Clique, Vertex-Cover Colouring-problem Hamilton-Cycle, Hamilton-Path

29 4:27 Theorems Walter Unger :00 SS2016 Z Example (Independet Set) Let G = (V, E) be a graph with pw(g) = k and V = n. Let P n k = ({p 1, p 2,..., p n k+1 }, {{p j, p j+1 } 1 j n k}) be a path with n k + 1 nodes. Let f : {p 1, p 2,..., p n k+1 } P(V ) the embedding function with pathwidth k. For each subset I j i I j i w j i f (p i ) store: and w j i with = I the size of the largest independet set I on i I j i I. Iteration step on f (p i+1 ): For each subset I j i+1 compute the above value: No there is no fun in this ugly task We have to simplify. j=1 f (p j) with

30 4:28 Theorems Walter Unger :00 SS2016 Z Example (Independet Set) Let G = (V, E) be a graph with pw(g) = k and V = n. Let P n k = ({p 1, p 2,..., p n k+1 }, {{p j, p j+1 } 1 j n k}) be a path with n k + 1 nodes. Let f : {p 1, p 2,..., p n k+1 } P(V ) the embedding function with pathwidth k with: f (p i ) f (p i+1 ) = 1, f (p i ) {x} = f (p i+1 ) or f (p i ) = f (p i+1 ) {x} Notations: f (p i+1 ) = add(f (p i ), x) and f (p i+1 ) = del(f (p i ), x)

31 4:29 Theorems Walter Unger :00 SS2016 Z Example (Independet Set) Thus we only have to define the following: What we store for f (p i ): D(f (p i )) The procedure D(f (p 1)) := Init(f (p 1)), to compute the initial values The procedure D(f (p i+1 )) := Add(D(f (p i )), x), to compute the values for f (p i+1 ). The procedure D(f (p i+1 )) := Del(D(f (p i )), x), to compute the values for f (p i+1 ).

32 4:30 Theorems Walter Unger :00 SS2016 Z Example (Independet Set) D(f (p i )) = {(I, w) I is IS on f (p i ) w = Wert(I, i)} with: Wert(I, i) = max{ I I i j=1f (p j ) I ist IS I I } Init(f (p 1)) = {(I, w) I is IS on f (p 1) w = I }, compute all IS I f (p 1) and set w = I. (I, w) Del(D(f (p i )), x) iff: (I {x}, w ) D(f (p i )) or (I, w ) D(f (p i )) and w = max{w (I {x}, w ) D(f (p i )) or (I, w ) D(f (p i ))}. (I, w) Add(D(f (p i )), x) iff: (I, w) D(f (p i )) or (I \ {x}, w 1) D(f (p i )) and I is IS.

33 4:31 Definition Walter Unger :00 SS2016 Z Treewidth Definition A graph G = (V, E) has treewidth k, iff there is a tree T = (V T, E T ) and a mapping f : V T P(V ) with: (v, u) E : x V T : v, u f (x) If c is on the path from a to b on T, then does f (b) f (a) f (c) hold. x V T : f (x) k + 1. and for k 1 exists no such function f and tree T. Notation: pw(g) = k. Note: T, f is called tree decomposition of width k.

34 4:32 Definition Walter Unger :00 SS2016 Z Example (Pathwidth and Treewidth) Σ =

35 4:33 Definition Walter Unger :00 SS2016 Z Example (Pathwidth and Treewidth) Σ =

36 4:34 Theorems Walter Unger :00 SS2016 Z Theorems Theorem Let G = (V, E) be a graph. Then the following hold: Theorem The problem, to compute the treewidth of a graph, is NP-complete. For a fixed k it is possible to check in linear time O(n + m), if a graph has treewidth k. Let G = (V, E) be a graphs with tw(g) = k. The following problem may be solved in linear time: Independent-Set, Clique, Vertex-Cover, k-dominating Set, Colouring-problem, Edge-Colouring, Hamilton-Cycle, Hamilton-Path, Graph-Isomorphie, Is-A-Disk-Graph-Problem,

37 4:35 Theorems Walter Unger :00 SS2016 Z Simplifications Let t be a successor of s in the tree. Then we may assume w.l.o.g.: s has at most two successors f (t) = f (s) holds if there is a second successor of s. f (t) f (s) = 1 if there is no second successor of s: f (s) = add(f (t), x) and f (s) = del(f (t), x)

38 4:36 Theorems Walter Unger :00 SS2016 Z Example (Independet Set) Thus we only have to define the following: What we store for f (p i ): D(f (p i )). The procedure D(f (p 1)) := Init(f (p 1)), to compute the initial values. The procedure D(f (p i+1 )) := Add(D(f (p i )), x), to compute the values for f (p i+1 ). The procedure D(f (p i+1 )) := Del(D(f (p i )), x), to compute the values for f (p i+1 ). The procedure D(f (s)) := Join(D(f (t)), D(f (t ))).

39 4:37 Theorems Walter Unger :00 SS2016 Z Example (Independet Set) D(f (s)) = {(I, w) I is IS on f (s) w = value(i, s)} with: value(i, s) = max{ I I t V (Ts )f (t) I is IS I I } and T s is the subtree with root s. Init(f (t)) = {(I, w) I is IS on f (t) w = I }, compute all IS I f (t) and set w = I. (I, w) Del(D(f (t)), x) iff: (I {x}, w ) D(f (p i )) or (I, w ) D(f (p i )) and w = max{w (I {x}, w ) D(f (p i )) or (I, w ) D(f (p i ))}. (I, w) Add(D(f (t)), x) iff: (I, w) D(f (t)) or (I \ {x}, w 1) D(f (t)) and I is IS. (I, w) Join(D(f (t)), D(f (t ))) iff: (I, w ) D(f (t)) and (I, w ) D(f (t )) and w = w + w I

40 4:38 Vertex Cover Walter Unger :00 SS2016 Z Vertex Cover and Treewidth Definition (Vertex Cover) Let G = (V, E) be a graph. The size of the minimal vertex cover is: vc(g) = min C C V : e E:e C Theorem Let G = (V, E) be a graph. Then tw(g) vc(g) holds. Theorem Let G = (V, E) be a graph. Then pw(g) vc(g) hold.

41 4:39 Vertex Cover Walter Unger :00 SS2016 Z Vertex Cover and Treewidth Theorem Let G = (V, E) be a graph. Then pw(g) vc(g) hold. Proof: Let C V with: e E : e C and C = k = vc(g). Let w.l.o.g. C = {v 1, v 2,..., v k } and V = {v 1, v 2,..., v n}. Furthermore let P = ({p k+1, p k+2,..., p n}, {{p j, p j+1 } k + 1 j < n}) be a path with n k nodes. Define f (p j ) = C {v j } for k + 1 j n. Then we have: f (p j ) vc(g) + 1 for k + 1 j n. {v c, v j } E and {v c, v j } C, then we get {v c, v j } f (p n). {v c, v j } E and {v c, v j } C = v c, then we get {v c, v j } f (p j ). Thus pw(g) vc(g) holds.

42 4:40 Example Walter Unger :00 SS2016 Z Idea (1-Tree) v1 Σ = 0 v1v2 v2 v3 v2v5 v2v4 v1v3 v4 v5 v6 v7 v5v10 v5v9 v4v8 v3v6 v3v12 v3v7 v6v11 v8 v9 v10 v11 v12 Σ = 0

43 4:41 Example Walter Unger :00 SS2016 Z Idea (2-Tree) v1 v2 Σ = 0 v1v2v3 v8 v4 v3 v6 v1v2v4 v1v3v5 v2v3v6 v5 v7 v1v5v8 v3v6v7 v9 v5v8v11 v6v7v9 v3v7v12 Σ = 0 v11 v12 v10 v7v9v10

44 4:42 Example Walter Unger :00 SS2016 Z Idea (3-Tree) v8 v3 v2 v12 Σ = 0 v1v2v3v4 v4 v1 v6 v5 v1v3v4v8 v1v3v4v7 v1v3v7v10 v1v4v7v9 v1v2v3v5 v1v2v5v6 v9 Σ = 0 v7 v10 v11 v1v2v6v11 v1v2v5v12

45 4:43 Definition Walter Unger :00 SS2016 Z k-tree Definition (k-tree (Rose 1974)) A k-tree is as follows recursively defined: K k+1 is a k-tree. Note: One may also start with K k. If T = (V, E) is a k-tree and C = {c 1, c 2,..., c k } is a clique in T, then is T = (V {v}, E {(v, c i ); 1 i k}) also a k-tree. There are no further k-trees. Let T = (V, E) be a k-tree. Then is G = (V, F ) with F E called a partial k-tree.

46 4:44 Theorems Walter Unger :00 SS2016 Z Theorems I Theorem A 1-tree is a tree. Let G be a k-tree. Then ω(g) = k + 1 holds if G has more than k nodes (otherwise ω(g) = k). ω(g) = max{ C C V (G) C ist Clique} Lemma A k-tree could be constructed by starting from any clique. Note A k-tree is chordal and perfect.

47 4:45 Theorems Walter Unger :00 SS2016 Z Theorems II Theorem A graph G = (V, E) is a k-tree, iff tw(g) = k and G is maximal. Theorem A graph G = (V, E) is a partial k-tree, iff tw(g) k.

48 4:46 Theorems Walter Unger :00 SS2016 Z Finding the Treewidth of a Graph It is hard to find the tree. One may use the following model: Modify the search-number. How many policeman are needed to find a person in a graph. The person may be arbitrary fast and may use nodes and edges. Policeman are only allowed to use the nodes, but may jump. The person may not pass a node where a policeman is. The policeman know the position of the person. An edge is called free (searched) if there is a policeman on both the incident nodes. Modified search-number corresponds to treewidth of a graph.

49 4:47 Cactus and near-trees Walter Unger :00 SS2016 Z Theorems III Definition A graph G = (V, E) is called cactus, iff each 2-connected component is a cycle. Theorem For a cactus G = (V, E) holds: tw(g) 2 Definition A graph G = (V, E) is called near-tree(k), iff each 2-connected component with x nodes has at most x + k 1 edges. Theorem For a near-tree(k) G = (V, E) holds: tw(g) k + 1

50 4:48 Cactus and near-trees Walter Unger :00 SS2016 Z Idea Cactus e3 e3c2r2 c2e3 r2e3 c2 r2 c2a0c0 r2 c2c0 c 2 c2e0r1 c 2 a1c2a0 c1c2c0 e1c2e0 r1c2 a1 c1 e1 r1 Σ = 0 a0 c0 e0 a1a0 c1c0 e1e0 r1 a0a1 c0c1 e0e1 Σ = 0 a0 c0 e0

51 4:49 Cactus and near-trees Walter Unger :00 SS2016 Z Proof (Cactus) Let G = (V, E) be a cactus with V = {v 1, v 2, v 3,..., v n}. Let C 1, C 2,..., C d be all cycles in G. Delete from each cycle C i one edge e i. Then T = (V, E \ {e 1, e 2,..., e d }) is a tree with root v 1. Modify now T as follows: For each node v define f (v) = {v}. For each edge {a, b} E(T ) generate a new node v a,b and define f (v a,b ) = {a, b}. Replace each edge {a, b} E(T ) by {a, v a,b }, {v a,b, b}. Replace each node v with deg(v) > 3 by a tree of degree 3. For all nodes x which replace v define f (x) = {v}. For each edge e i = {a, b} let x be the smallest common predecessor. For each node y on the path from a to x define f (y) = f (y) {a}. For each node y on the path from b to x define f (y) = f (y) {b}. We have for each node z: f (z) 3.

52 4:50 Cactus and near-trees Walter Unger :00 SS2016 Z Idea near-tree e3 e3c2 c2e3 r2e3c2 c2 r2 c2 r2c2 cc2 c2 c2a1a0 cc1 c2e0 ce1 a1c2a0 c1c2a1a0 e1c2e0 r1c2e0 a1 c1 e1 r1 Σ = 0 a0 c0 e0 a1a0 c1a1a0 e1e0 r1e0 a0a1 c0c1a0 e0e1 Σ = 0 a0 c0a0 e0

53 4:51 Cactus and near-trees Walter Unger :00 SS2016 Z Proof (near-tree) Let G = (V, E) be a near-tree(k) with V = {v 1, v 2, v 3,..., v n}. Let C 1, C 2,..., C d be all 2-connected components in G. Delete from each component C i up to d i k edges e j i. Then is T = (V, E \ {e j i 1 i d 1 j d i }) a tree with root v 1. Modify now T as follows: For each node v define f (v) = {v}. Replace any node v which is several 2-connected components by a star. For all nodes x which replace v define f (x) = {v}. For each edge {a, b} E(T ) generate a new node v a,b and define f (v a,b ) = {a, b}. Replace each edge {a, b} E(T ) by {a, v a,b }, {v a,b, b}. For each edge e j i = {a, b} and for each node y on the path from a to b define f (y) = f (y) {a}. We have for each node z: f (z) 2 + k.

54 4:52 Halin-graphs and outer-planar graphs Walter Unger :00 SS2016 Z Theorems IV Definition A graph G = (V, E) is called Halin-graph, iff G is a planar embedded tree where the leave are connected by the cycle. Theorem For a Halin-Graph G = (V, E) holds: tw(g) 3 Definition A planar graph G = (V, E) is called outer-planar, iff it could be drawn in the plane, such that no two edges cross and all nodes are on the outer window. Theorem For a outer-planar graph G = (V, E) holds: tw(g) 2

55 4:53 Halin-graphs and outer-planar graphs Walter Unger :00 SS2016 Z Idea Halin e3 e3r1r2 c2e3r1r2 r2e3r1 c2 r2 c2a0r1r2 r2r1 c2a0c0r1 c 2 c2c0e0r1c 2 a1c2a0r2 c1c2a0c0 e1c2c0e0 r1c2e0 a1 c1 e1 r1 Σ = 0 a0 c0 e0 a1a0r2 c1a0c0 e1c0e0 r1e0 a0a1r2 c0c1a0 e0e1c0 Σ = 0 a0r2 c0a0 e0c0

56 4:54 Halin-graphs and outer-planar graphs Walter Unger :00 SS2016 Z Proof I (Halin Graph) Let G = (V, E) be a Halin-graph with V = {v 1, v 2, v 3,..., v n}. Let T = (V, E ) be the tree of G with root v 1. Let (a 1, a 2,..., a k ) be the cycle connecting the leaves. Modify now T as follows: For each node v define f (v) = {v}. For each edge {a, b} E(T ) generate a new node v a,b and define f (v a,b ) = {a, b}. Replace each edge {a, b} E(T ) by {a, v a,b }, {v a,b, b}. Replace each node v with deg(v) > 3 by a tree of degree 3. For all nodes x which replace v define f (x) = {v}. For each edge {a i, a i+1 } and for each node y on the path from a i to a i+1 define f (y) = f (y) {a i }. For edge {a k, a 1} and for each node y on the path from a k to a 1 define f (y) = f (y) {a k }. We have for each node z: f (z) 4.

57 4:55 Halin-graphs and outer-planar graphs Walter Unger :00 SS2016 Z Idea outer-planar Graph v6 v4v5v6 v2v4v5 v4 v5 v1v2v4 Σ = 0 v2v3v5 Σ = 0 v1 v2 v3 v2v3v7 v7

58 4:56 Halin-graphs and outer-planar graphs Walter Unger :00 SS2016 Z Proof II (Outer-planar Graph) Let G = (V, E) be a outer-planar graph. Let G = (V, E) be maximal, i.e. G = (V, E {e}) is not outer-planar. Let V be the set of inner regions (windows). Then is V a set of triangles. For x V let: V (x) be the nodes of triangle x. E(x) be the edges of triangle x. Define tree (V, {{a, b} E(a) E(b) }) Define f by f (x) = V (x).

59 4:57 Halin-graphs and outer-planar graphs Walter Unger :00 SS2016 Z Theorems V Definition A planar graph G = (V, E) is called k-outer-planar, iff there is a planar embedding of G such that after deleting k 1 times all nodes of the outer window the remaining graph embedded as an outer-planar graph. Theorem For k-outer-planar graphs G = (V, E) holds: tw(g) 3 k 1

60 4:58 SP-Graphs Walter Unger :00 SS2016 Z Theorems VI Definition A graph G = (V, E) is called SP-graph (series-parallel graph), iff it may be constructed by using series-parallel operations: G = ({a, b},, a, b) is a SP-graph. G = ({a, b}, {{a, b}}, a, b) is a SP-graph. If G = (V, E, a, b) and G = (V, E, a, b) are SP-graphs with V V = {a, b} then G = (V V, E E, a, b) is a SP-graph. If G = (V, E, a, x) and G = (V, E, x, b) are SP-graphs with V V = {x} then G = (V V, E E, a, b) is a SP-graph. Theorem A SP-graph has treewidth 2.

61 4:59 Minors Walter Unger :00 SS2016 Z Minors Definition (Minor) A graph G is the minor of a graph G, iff an isomorphic image of G could be generated from G by node-merging of connected nodes. Merging of nodes: Let G = (V, E) Let {a, b} E, Then the node-merging of a and b is possible: G = (V \ b, (E \ {{v, b} v V }) {{v, a} {v, b} E})

62 4:60 Minors Walter Unger :00 SS2016 Z Theorems VII Theorem A G with tw(g) k has no K k+2 minor. Theorem Graphs G with tw(g) k could be described by a bounded sequence of minors. Theorem Any problem described in MS 2 on a graph G with tw(g) k is solvable in polynomial time.

63 5 Inhaltsverzeichnis Walter Unger :00 SS2016 Z Questions 1 What is the definition of bandwidth? 2 Which problems may be solved on graphs with bounded bandwidth? 3 What is the idea for this? 4 What is the definition of pathwidth? 5 Which problems may be solved on graphs with bounded pathwidth? 6 What is the idea for this? 7 Compare the bandwidth and the pathwidth of a graph. 8 What is the definition of treewidth? 9 Which problems may be solved on graphs with bounded treewidth? 10 What is the idea for this?

64 5 Inhaltsverzeichnis Walter Unger :00 SS2016 Z Questions 1 Compare the treewidth and the pathwidth of a graph. 2 What is the definition of a partial k-tree? 3 Compare treewidth and partial k-tree. 4 What is the treewidth of a Halin-graph? 5 What is the treewidth of a cactus? 6 What is the treewidth of a near-tree(k)? 7 What is the treewidth of an outer-planar graphen? 8 What is the treewidth of a k-outer-planar Graphen? 9 What is the treewidth of a SP-graph?

65 5 Inhaltsverzeichnis Walter Unger :00 SS2016 Z Legend : Not of relevance : implicitly used basics : idea of proof or algorithm : structure of proof or algorithm : Full knowledge