TREE-DECOMPOSITIONS OF GRAPHS. Robin Thomas. School of Mathematics Georgia Institute of Technology thomas

Transcription

1 1 TREE-DECOMPOSITIONS OF GRAPHS Robin Thomas School of Mathematics Georgia Institute of Technology thomas

2 MOTIVATION 2 δx = edges with one end in X, one in V (G) X

3 MOTIVATION 3 δx = edges with one end in X, one in V (G) X

4 MOTIVATION 4 δx = edges with one end in X, one in V (G) X Two edge-cuts δx, δy do not cross if: X Y or X Y c or X c Y or X c Y c.

5 Example of a cross-free family of edge-cuts: 5 Let T be a tree, and (W t : t V (T )) a partition of V (G). Every edge of T defines a cut; the collection of cuts thus obtained is cross-free.

6 A separation of a graph G is a pair (A, B) such that A B = V (G) and there is no edge between A B and B A. 6

7 A separation of a graph G is a pair (A, B) such that A B = V (G) and there is no edge between A B and B A. 7

8 A separation of a graph G is a pair (A, B) such that A B = V (G) and there is no edge between A B and B A. 8

9 A separation of a graph G is a pair (A, B) such that A B = V (G) and there is no edge between A B and B A. 9

10 A separation of a graph G is a pair (A, B) such that A B = V (G) and there is no edge between A B and B A. 10 Two separations (A, B) and (C, D) do not cross if: A C and B D, or A D and B C, or A C and B D, or A D and B C.

11 A separation of a graph G is a pair (A, B) such that A B = V (G) and there is no edge between A B and B A. 11 Two separations (A, B) and (C, D) do not cross if: A C and B D, or A D and B C, or A C and B D, or A D and B C.

12 A family of cross-free separations gives rise to a tree-decompositon. 12

13 A tree-decomposition of a graph G is (T, W ), where T is a tree and W = (W t : t V (T )) satisfies (T1) t V (T ) W t = V (G), (T2) if t T [t, t ], then W t W t W t, (T3) uv E(G) t V (T ) s.t. u, v W t. 13 The width is max( W t 1 : t V (T )). The tree-width of G is the minimum width of a tree-decomposition of G.

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15 tw(g) 1 G is a forest 15

16 tw(g) 1 G is a forest 15 tw(g) 2 G is series-parallel

17 tw(g) 1 G is a forest 15 tw(g) 2 G is series-parallel tw(g) 3 no minor isomorphic to: K 5, 5-prism, octahedron, V 8

18 tw(g) 1 G is a forest 15 tw(g) 2 G is series-parallel tw(g) 3 no minor isomorphic to: K 5, 5-prism, octahedron, V 8 tw(k n ) = n 1

19 tw(g) 1 G is a forest 15 tw(g) 2 G is series-parallel tw(g) 3 no minor isomorphic to: K 5, 5-prism, octahedron, V 8 tw(k n ) = n 1 tree-width is minor-monotone

20 tw(g) 1 G is a forest 15 tw(g) 2 G is series-parallel tw(g) 3 no minor isomorphic to: K 5, 5-prism, octahedron, V 8 tw(k n ) = n 1 tree-width is minor-monotone The k k grid has tree-width k

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22 Consider all functions φ mapping graphs into integers such that 17 (1) φ(k n ) = n 1, (2) G minor of H φ(g) φ(h), (3) If G H is a clique, then φ(g H) = max{φ(g), φ(h)}. Order such functions by φ ψ if φ(g) ψ(g) for all G. THEOREM (Halin) Tree-width is the maximum element in the above poset.

23 A haven β of order k in G assigns to every X [V (G)] <k the vertex-set of a component of G\X such that 18 (H) X Y [V (G)] <k β(y ) β(x).

24 A haven β of order k in G assigns to every X [V (G)] <k the vertex-set of a component of G\X such that 18 (H) X Y [V (G)] <k β(y ) β(x). X β(χ)

25 A haven β of order k in G assigns to every X [V (G)] <k the vertex-set of a component of G\X such that 19 (H) X Y [V (G)] <k β(y ) β(x). Υ X β(χ)

26 A haven β of order k in G assigns to every X [V (G)] <k the vertex-set of a component of G\X such that 20 (H) X Y [V (G)] <k β(y ) β(x). Υ X β(χ)

27 A haven β of order k in G assigns to every X [V (G)] <k the vertex-set of a component of G\X such that 21 (H) X Y [V (G)] <k β(y ) β(x). Υ β(υ) X β(χ)

28 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 22

29 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 22.

30 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 23.

31 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 24.

32 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 25.

33 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 26.

34 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 27.

35 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 28.

36 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 29.

37 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 30.

38 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 31.

39 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 32

40 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 32 Fact. A tree-decomposition of width k 1 gives a search strategy for k cops.

41 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 32 Fact. A tree-decomposition of width k 1 gives a search strategy for k cops. Fact. A haven gives an escape strategy for the robber.

42 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 32 Fact. A tree-decomposition of width k 1 gives a search strategy for k cops. Fact. A haven gives an escape strategy for the robber. THEOREM (Seymour, RT) G has a haven of order k G has tree-with at least k 1

43 Cops and robbers. Fix a graph G and an integer k. There are k cops, they move slowly in helicopters. There is a robber, who moves infinitely fast along cop-free paths. He can see a helicopter landing, and can run to a safe place before the chopper lands. 32 Fact. A tree-decomposition of width k 1 gives a search strategy for k cops. Fact. A haven gives an escape strategy for the robber. THEOREM (Seymour, RT) G has a haven of order k G has tree-with at least k 1 COR Search strategy monotone search strategy.

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45 THEOREM (Robertson, Seymour, RT) Every graph of tree-width 20 2g5 has a g g grid minor. 34

46 THEOREM (Robertson, Seymour, RT) Every graph of tree-width 20 2g5 has a g g grid minor. 34 THEOREM (Bodlaender) For every k there is a linear-time algorithm to decide whether tw(g) k.

47 THEOREM (Robertson, Seymour, RT) Every graph of tree-width 20 2g5 has a g g grid minor. 34 THEOREM (Bodlaender) For every k there is a linear-time algorithm to decide whether tw(g) k. THEOREM (Arnborg, Proskurowski,...) Many problems can be solved in linear time when restricted to graphs of bounded tree-width.

48 Tree-width is useful in 35 theory design of theoretically fast algorithms practical computations

49 FEEDBACK VERTEX-SET FOR FIXED k 36 INSTANCE A graph G QUESTION Is there a set X V (G) such that X k and G\X is acyclic? ALGORITHM If tw(g) is small use bounded tree-width methods. Otherwise answer no. That s correct, because big tree-width big grid k + 1 disjoint circuits X does not exist.

50 k DISJOINT PATHS IN PLANAR GRAPHS 37 INSTANCE A planar graph G, vertices s 1, s 2,..., s k, t 1, t 2,..., t k of G QUESTION Are there disjoint paths P 1,.., P k such that P i has ends s i and t i? s 1 t 1 s s s t t t 2 3 4

51 k DISJOINT PATHS IN PLANAR GRAPHS 38 INSTANCE A planar graph G, vertices s 1, s 2,..., s k, t 1, t 2,..., t k of G QUESTION Are there disjoint paths P 1,.., P k such that P i has ends s i and t i?

52 k DISJOINT PATHS IN PLANAR GRAPHS 38 INSTANCE A planar graph G, vertices s 1, s 2,..., s k, t 1, t 2,..., t k of G QUESTION Are there disjoint paths P 1,.., P k such that P i has ends s i and t i? ALGORITHM tw(g) small bounded tree-width methods. Otherwise big grid minor big grid minor with the terminals outside. The middle vertex of this grid minor can be deleted, without affecting the feasibility of the problem.

53 APPLICATIONS 39 THEOREM (Erdös, Pósa) There exists a function f such that every graph has either k disjoint cycles, or a set X of at most f(k) vertices such that G\X is acyclic. THEOREM (Robertson, Seymour) For every planar graph H there exists a function f such that every graph has either k disjoint H minors, or a set X of at most f(k) vertices such that G\X has no H minor. False for every nonplanar graph H. Open for subdivisions.

54 THEOREM (Oporowski, Oxley, RT) There exists a function f such that every 3-connected graph on at least f(t) vertices has a minor isomorphic to W t or K 3,t. 40 THEOREM (Oporowski, Oxley, RT) There exists a function f such that every 4-connected graph on at least f(t) vertices has a minor isomorphic to D t, M t, O t, or K 4,t. THEOREM (Ding, Oporowski, RT, Vertigan) There exists a function f such that every 4-connected nonplanar graph on at least f(t) vertices has a minor isomorphic to D t, M t, or K 4,t.

55 COROLLARY (Ding, Oporowski, RT, Vertigan) There exists a constant c such that every minimal graph of crossing number at least two on at least c vertices belongs to a well-defined family of graphs. 41

56 THEOREM (Arnborg, Proskurowski) Let P (G, Z) be some information about a graph G and set Z V (G) such that (i) P (G, Z) can be computed in constant time if V (G) k + 1 (ii) if Z Z then P (G, Z ) can be computed from P (G, Z) in constant time (iii) if (A, B) is a separation of G with A B Z, then P (G, Z) can be computed from P (G A, A Z), P (G B, B Z) in constant time. 42 Then P (G, ) can be computed in linear time if a tree-decomposition of G of width k is given.

57 EXAMPLE. For A V (G), let α A be the maximum cardinality of an independent set I V (G) with I Z = A. Let P (G, Z) = (α A : A Z). 43

58 Discrete magnetic Schrödinger operators 44 M G = all Hermitian matrices A = (a ij ) s.t. a i,j 0 if i, j are adjacent and a i,j = 0 if i j and not adjacent.

59 Discrete magnetic Schrödinger operators 44 M G = all Hermitian matrices A = (a ij ) s.t. a i,j 0 if i, j are adjacent and a i,j = 0 if i j and not adjacent. W l = all positive semi-definite Hermitian matrices with l-dimensional kernel

60 Discrete magnetic Schrödinger operators 44 M G = all Hermitian matrices A = (a ij ) s.t. a i,j 0 if i, j are adjacent and a i,j = 0 if i j and not adjacent. W l = all positive semi-definite Hermitian matrices with l-dimensional kernel DEFINITION (Colin de Verdière) Let ν(g) be the maximum l such that there exists A M G W l such that those manifolds intersect transversally at A.

61 Discrete magnetic Schrödinger operators 44 M G = all Hermitian matrices A = (a ij ) s.t. a i,j 0 if i, j are adjacent and a i,j = 0 if i j and not adjacent. W l = all positive semi-definite Hermitian matrices with l-dimensional kernel DEFINITION (Colin de Verdière) Let ν(g) be the maximum l such that there exists A M G W l such that those manifolds intersect transversally at A. THEOREM (Colin de Verdière) ν(g) tw (G), where tw (G) is a slight variation of tree-width s.t. tw(g) tw (G) tw(g) + 1.

62 A path decomposition of G is a sequence W 1, W 2,..., W n such that 45 (i) W i = V (G), and every edge has both ends in some W i, and (ii) if i < i < i then W i W i W i The width of W 1,..., W n is max{ W i 1 : 1 i n} The path-width of G is the minimum width of a path-decomposition.

63 THM F forest, pw(g) V (F ) 1 F m G. 46

64 THM F forest, pw(g) V (F ) 1 F m G. 46 HISTORY Originally due to Robertson and Seymour. Current bound by Bienstock, Robertson, Seymour, RT. New proof by Diestel.

65 THM F forest, pw(g) V (F ) 1 F m G. 47

66 THM F forest, pw(g) V (F ) 1 F m G. 47 Diestel s proof. Let V (F ) = {v 1, v 2,..., v k } s.t. v i is adjacent to 1 v j for j < i. Let L = {(A, B) : G B has no path-decomposition W 1, W 2,..., W t of width k 2 with A B W 1 }. Choose i {0, 1,..., k} and (A, B) L such that (i) G A has a minor isomorphic to F {v 1,..., v i } s.t. each node intersects A B in precisely one vertex (ii) (A, B ) L with A A, B B, A B < A B (iii) i is maximum subject to (i) and (ii) (iv) B is minimum subject to (i), (ii), (iii)

67 CLAIM (A, B ) L with A A, B B, A B A B. 48

68 PROOF OF CLAIM. Suppose not. By (ii) equality holds. 49

69 PROOF OF THM Let j be the only index i such that v i+1 v j. Pick any vertex of B A adjacent to the unique vertex of A B that belongs to the v j -node. 50

70 Let (T, W ) be a tree-decomposition of G and t V (T ). The torso at t is G W t plus all edges with both ends in W t W t for some t t. 51 (T, W ) is a tree-decomposion over F if every torso belongs to F.

71 HOW TO USE A HAVEN? 52 REMINDER A haven β of order k in D assigns to every X [V (D)] <k the vertex-set of a strong component of D\X such that (H) X Y [V (D)] <k β(y ) β(x).

72 Let β be a haven of order k in G. Let X V (G) with X k/2 and β(x) minimum. Then X is externally linked : 53 X β(χ)

73 Let β be a haven of order k in G. Let X V (G) with X k/2 and β(x) minimum. Then X is externally linked : 54 X β(χ)

74 Let β be a haven of order k in G. Let X V (G) with X k/2 and β(x) minimum. Then X is externally linked : 55 X β(χ)

75 Let β be a haven of order k in G. Let X V (G) with X k/2 and β(x) minimum. Then X is externally linked : 56 X Z

76 Let β be a haven of order k in G. Let X V (G) with X k/2 and β(x) minimum. Then X is externally linked : 57 X Z

77 Let β be a haven of order k in G. Let X V (G) with X k/2 and β(x) minimum. Then X is externally linked : 58 β(χ+ζ) X Z

78 Let β be a haven of order k in G. Let X V (G) with X k/2 and β(x) minimum. Then X is externally linked : 59 β(χ+ζ)=β(υ) X Z

79 Let Σ be a surface with k holes, C 1,..., C k their boundaries ( cuffs ). 60 A graph G can be nearly embedded in Σ if G has a set X of at most k vertices such that G\X can be written as G 0 G 1... G k, where for i > 0: 1. G 0 has an embedding in Σ 2. G i are pairwise disjoint 3. U i := V (G 0 ) V (G i ) = V (G 0 ) C i 4. G i has a path decomposition (X u ) u Ui of width < k, s.t. u X u for all u U i (the order on U i given by C i ) NOTATION: G F(Σ)

80 Σ k = Σ with k holes removed 61 Σ H = orientable surface of largest genus that does not embed H Σ H same for non-orientable THEOREM (Robertson, Seymour) For every finite graph H there exists k 0 such that every graph with no H minor has a tree-decomposition over F(Σ H k) F(Σ H k).

81 INFINITE GRAPHS 62 THEOREM (Halin) A graph has no ray (= 1-way infinite path) it has a tree-decomposition (T, W ) such that T is rayless and each W t is finite. With Robertson and Seymour we characterize graphs with no K κ minor, no T κ subdivision, or no half-grid minor. Havens and searching play an important role. SAMPLE RESULT. A graph has no T ℵ1 -minor it has no tree-decomposition (T, W ), where T is rayless and each W t is at most countable.

82 MOTIVATION 63 THEOREM (RT) There exists a sequence G 1, G 2,... of (uncountable) graphs such that for i < j G j has no G i minor. CONJECTURE True for countable graphs. THEOREM (RT) Known when G 1 is finite and planar. FACT Not known even when every component is finite.

83 LEMMA (Kříž, RT) Let F be compact (if every finite subgraph of G belongs to F, then G F). If every finite subgraph of G has a tree-decomposition over F, then so does G. 64

84 THEOREM (Diestel, Thomas) For every finite graph H there exists an integer k such that every (infinite) graph with no H minor has a tree-decomposition over F(Σ H k) F(Σ H k). 65

85 A graph G is plane with one vortex if for some k it has a near-embedding G 0, G 1,.., G k in the sphere with k holes, where G 2,..., G k are null. 66 A tree-dec. (T, W ) has finite adhesion if for every t, W t W t is bounded (t t), for every ray t 1, t 2,... in T, lim inf W ti W ti+1 is finite. THEOREM (Diestel, Thomas) An infinite graph has no K ℵ0 -minor if and only if it has a tree-decomposition of finite adhesion over plane graphs with at most one vortex.

86 THEOREM (Robertson, Seymour, RT) 67 Every planar graph with no minor isomorphic to a g g grid has tree-width < 5g. PROOF Suppose G has tree-width 5g. Then G has a haven β of order 5g. Take a planar drawing of G and a circular cutset X of order 4g with β(x) inside X and with inside of X minimal.