Cuts and trees. Bernhard Krön. University of Vienna. joint with M.J. Dunwoody (Univ. of Southampton, UK) Friday, 21. Nov. 2008
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1 Cuts and trees Bernhard Krön University of Vienna joint with M.J. Dunwoody (Univ. of Southampton, UK) Friday, 21. Nov Bernhard Krön (Uni Vienna) Vertex cuts / 27
2 Contents 1 Background 1: tree decompositions 2 Ends of graphs and groups 3 Background 2: Seifert - van Kampen and Stallings structure theorem 4 Axiomatic definition of cut-systems 5 The main result 6 Applications Bernhard Krön (Uni Vienna) Vertex cuts / 27
3 1- and 2-connected graphs X 1-connected, tree decomosition: cut-points and 2-connected blocks. Bernhard Krön (Uni Vienna) Vertex cuts / 27
4 1- and 2-connected graphs X 1-connected, tree decomosition: cut-points and 2-connected blocks. small = one element, large = 2-connected blocks Bernhard Krön (Uni Vienna) Vertex cuts / 27
5 1- and 2-connected graphs X 1-connected, tree decomosition: cut-points and 2-connected blocks. small = one element, large = 2-connected blocks X 2-connected, tree decomposition similar (Tutte, Droms, Servatius, Servatius). small = two element, large = 3-connected blocks Bernhard Krön (Uni Vienna) Vertex cuts / 27
6 1- and 2-connected graphs X 1-connected, tree decomosition: cut-points and 2-connected blocks. small = one element, large = 2-connected blocks X 2-connected, tree decomposition similar (Tutte, Droms, Servatius, Servatius). small = two element, large = 3-connected blocks We cannot decompose 3-connected graphs into 4-connected blocks like that. Bernhard Krön (Uni Vienna) Vertex cuts / 27
7 tree decomposition tree Bernhard Krön (Uni Vienna) Vertex cuts / 27
8 ends of graphs graph X = (VX, EX ) ray R R 1 R 2 R 3 such that R 3 R 1 = R 3 R 2 =. Bernhard Krön (Uni Vienna) Vertex cuts / 27
9 ends of graphs graph X = (VX, EX ) ray R R 1 R 2 R 3 such that R 3 R 1 = R 3 R 2 =. Equivalence classes are the ends of X. This definition is due to Halin (1964). Bernhard Krön (Uni Vienna) Vertex cuts / 27
10 C VX, NC = {x VX \ C x C}, C = VX \ (C NC) write R C if C \ R <. Bernhard Krön (Uni Vienna) Vertex cuts / 27
11 C VX, NC = {x VX \ C x C}, C = VX \ (C NC) write R C if C \ R <. Equivalent definitition: R 1 R 2 if C s.t. NC <, R 1 C and R 2 C That is, large sets C and C are separated by the small set NC. small = finite set of vertices, large = containing a ray (i.e., an end) Bernhard Krön (Uni Vienna) Vertex cuts / 27
12 C VX, NC = {x VX \ C x C}, C = VX \ (C NC) write R C if C \ R <. Equivalent definitition: R 1 R 2 if C s.t. NC <, R 1 C and R 2 C That is, large sets C and C are separated by the small set NC. small = finite set of vertices, large = containing a ray (i.e., an end) Freudenthal s original definition from 1931 for locally compact, connected space with a countable base: small = compact, large = not compact, open, connected In 1945 he studied ends of locally finite graphs. Bernhard Krön (Uni Vienna) Vertex cuts / 27
13 Cayley graphs G = S finitely generated, X = Cay(G, S) Cayley graph Bernhard Krön (Uni Vienna) Vertex cuts / 27
14 Cayley graphs G = S finitely generated, X = Cay(G, S) Cayley graph VX = G EX : x y s S such that xs = y or x y x 1 y S If S = S 1 = {s 1 s S} then X is undirected. Bernhard Krön (Uni Vienna) Vertex cuts / 27
15 free amalgamated products Bernhard Krön (Uni Vienna) Vertex cuts / 27
16 HNN-extensions Bernhard Krön (Uni Vienna) Vertex cuts / 27
17 Stallings structure theorem Bernhard Krön (Uni Vienna) Vertex cuts / 27
18 Stallings structure theorem G splits over H < G if G = G 1 α G 2 or G = G 1 α for some α : H H. Bernhard Krön (Uni Vienna) Vertex cuts / 27
19 Stallings structure theorem G splits over H < G if G = G 1 α G 2 or G = G 1 α for some α : H H. Theorem (Stallings structure theorem, 1971) G finitely generated. Then G has more than one end G splits over finite subgroup Bernhard Krön (Uni Vienna) Vertex cuts / 27
20 Dicks and Dunwoody s theory Combinatorial (graph theoretic) proof of Stallings theorem by Dunwoody: Bernhard Krön (Uni Vienna) Vertex cuts / 27
21 Dicks and Dunwoody s theory Combinatorial (graph theoretic) proof of Stallings theorem by Dunwoody: Automorphism-invariant tree decompositions where small = finite set of edges, large = containing a ray. Bernhard Krön (Uni Vienna) Vertex cuts / 27
22 Dicks and Dunwoody s theory Combinatorial (graph theoretic) proof of Stallings theorem by Dunwoody: Automorphism-invariant tree decompositions where small = finite set of edges, large = containing a ray. The crucial part of the proof is a complicated construction of certain nice edge-cuts (paper Cutting up graphs 1982), uses Bass-Serre theory. Bernhard Krön (Uni Vienna) Vertex cuts / 27
23 Dicks and Dunwoody s theory Combinatorial (graph theoretic) proof of Stallings theorem by Dunwoody: Automorphism-invariant tree decompositions where small = finite set of edges, large = containing a ray. The crucial part of the proof is a complicated construction of certain nice edge-cuts (paper Cutting up graphs 1982), uses Bass-Serre theory. Fully developed theory in the book M.J. Dunwoody, W. Dicks Groups acting on Graphs, Bernhard Krön (Uni Vienna) Vertex cuts / 27
24 historic root = Seifert - van Kampen theorem X = U V U U V V π 1 (X, x) = π 1 (U, x) π1 (U V,x) π 1 (V, x), x U V Bernhard Krön (Uni Vienna) Vertex cuts / 27
25 Axioms - vertex cuts Let X be a connected graph. Sets C and S of non-empty sets of vertices are called cut-system and separator-system, respectively, if they satisfy the following. Bernhard Krön (Uni Vienna) Vertex cuts / 27
26 Axioms - vertex cuts Let X be a connected graph. Sets C and S of non-empty sets of vertices are called cut-system and separator-system, respectively, if they satisfy the following. (A1) Separators are finite sets whose complements have at least two components which contain a cut. Bernhard Krön (Uni Vienna) Vertex cuts / 27
27 Axioms - vertex cuts Let X be a connected graph. Sets C and S of non-empty sets of vertices are called cut-system and separator-system, respectively, if they satisfy the following. (A1) Separators are finite sets whose complements have at least two components which contain a cut. (A2) Cuts are the components of the complement of a separator which contain a cut. Bernhard Krön (Uni Vienna) Vertex cuts / 27
28 Axioms - vertex cuts Let X be a connected graph. Sets C and S of non-empty sets of vertices are called cut-system and separator-system, respectively, if they satisfy the following. (A1) Separators are finite sets whose complements have at least two components which contain a cut. (A2) Cuts are the components of the complement of a separator which contain a cut. (A3) A cut minus a separator contains a cut. Bernhard Krön (Uni Vienna) Vertex cuts / 27
29 Axioms - vertex cuts Let X be a connected graph. Sets C and S of non-empty sets of vertices are called cut-system and separator-system, respectively, if they satisfy the following. (A1) Separators are finite sets whose complements have at least two components which contain a cut. (A2) Cuts are the components of the complement of a separator which contain a cut. (A3) A cut minus a separator contains a cut. Note: S is determined by C, but C is usually not determined by S. A set is called large if it contains a cut. Bernhard Krön (Uni Vienna) Vertex cuts / 27
30 Examples of cut systems Example X 1-connected S = 1-element separators (cut-points) C = components obtained by removing a separator Bernhard Krön (Uni Vienna) Vertex cuts / 27
31 Examples of cut systems for us the most important example: Example X connected S = finite sets which separate ends C = those components in the complement of a separator, which contain a ray (i.e., an end). Bernhard Krön (Uni Vienna) Vertex cuts / 27
32 Examples of cut systems more general: Example X connected, M VX S = finite subsets of M which separate ends C = components in the complement of a separator, which contain a ray. Bernhard Krön (Uni Vienna) Vertex cuts / 27
33 Examples of cut systems more general: Example X connected, M VX S = finite subsets of M which separate ends C = components in the complement of a separator, which contain a ray. The following is the setting in the book of Dicks and Dunwoody. Example edge cuts: put an additional vertex on each edge and let M be the set of these new vertices. Bernhard Krön (Uni Vienna) Vertex cuts / 27
34 vertex cuts - edge cuts E edge cut E vertex cut Bernhard Krön (Uni Vienna) Vertex cuts / 27
35 vertex cuts - edge cuts E edge cut E vertex cut E vertex cut E edge cut Bernhard Krön (Uni Vienna) Vertex cuts / 27
36 vertex cuts - edge cuts E edge cut E vertex cut E vertex cut E edge cut X locally finite: E vertex cut E edge cut Bernhard Krön (Uni Vienna) Vertex cuts / 27
37 What about finite graphs? Y VX is called k-inseparable if Y > k and for all S VX, S = k, the set Y is contained in the union of S and one component of VX \ S. In other words: No k-set S separates Y. Bernhard Krön (Uni Vienna) Vertex cuts / 27
38 What about finite graphs? Y VX is called k-inseparable if Y > k and for all S VX, S = k, the set Y is contained in the union of S and one component of VX \ S. In other words: No k-set S separates Y. Example Let κ be the minimal integer (if it exists) such that there is a κ-set S which separates two κ-inseparable sets Y 1 and Y 2 in the sense that there are components C 1 and C 2 of VX \ S, C 1 C 2 and Y 1 C 1 S and Y 2 C 2 S. S = κ-sets which separate two κ-inseparable sets C = those components in the complement a separator, which contain a κ-inseparable set. Bernhard Krön (Uni Vienna) Vertex cuts / 27
39 minimal cuts Let C be a cut system and let κ be the minimal cardinality of a separator. A cut whose boundary has κ elements is called a minimal cut. Bernhard Krön (Uni Vienna) Vertex cuts / 27
40 minimal cuts Let C be a cut system and let κ be the minimal cardinality of a separator. A cut whose boundary has κ elements is called a minimal cut. Theorem (Dunwoody, Krön) The set of minimal cuts is a cut system. Bernhard Krön (Uni Vienna) Vertex cuts / 27
41 nestedness 4 corners, 4 links, 1 center. Bernhard Krön (Uni Vienna) Vertex cuts / 27
42 nestedness 4 corners, 4 links, 1 center. isolated corner = not large and both adjacent links are empty Bernhard Krön (Uni Vienna) Vertex cuts / 27
43 nestedness 4 corners, 4 links, 1 center. isolated corner = not large and both adjacent links are empty Sets of vertices E and F are nested if they have an isolated corner. Bernhard Krön (Uni Vienna) Vertex cuts / 27
44 Main theorem Theorem (Dunwoody, Krön) Every cut system has a minimal cut which is nested with all other cuts. Bernhard Krön (Uni Vienna) Vertex cuts / 27
45 Main theorem Theorem (Dunwoody, Krön) Every cut system has a minimal cut which is nested with all other cuts. The existence of so-called structure cuts (or D-cuts) follows as corollary. See Dunwoody Cutting up graphs 1982, Dicks and Dunwoody Groups acting on Graphs Bernhard Krön (Uni Vienna) Vertex cuts / 27
46 Counterexample In the papers on edge cuts, nestedness of cuts E and F means that one of following inclusions hold E F or E VX \ F or VX \ E F or VX \ E VX \ F. Equivalently, E F or F E or E F = or E F = VX. There are graphs with more than one end, where there is no automorphism invariant cut system which satisfies the above condition for all pairs of cuts. We have to use our weaker concept of nestedness which uses isolated corners. Bernhard Krön (Uni Vienna) Vertex cuts / 27
47 Generalized structure trees Let G be a group acting on X and let C be a G-set. Let N be the set of minimal cuts which are nested with all other cuts. Then N is a G-set. And there is a tree T with ET = N and VT = S N / where is an equivalence relation on N. The equivalence classes correspond to the blocks between the separators. Bernhard Krön (Uni Vienna) Vertex cuts / 27
48 Generalized structure trees Let G be a group acting on X and let C be a G-set. Let N be the set of minimal cuts which are nested with all other cuts. Then N is a G-set. And there is a tree T with ET = N and VT = S N / where is an equivalence relation on N. The equivalence classes correspond to the blocks between the separators. Then G acts on T such that the stabilizers of the edges of T are the stabilizers of cuts in N. Bernhard Krön (Uni Vienna) Vertex cuts / 27
49 cuts tree Bernhard Krön (Uni Vienna) Vertex cuts / 27
50 Why two types of vertices? Bernhard Kro n (Uni Vienna) Vertex cuts / 27
51 Generalization of Stallings Structure Theorem Bernhard Krön (Uni Vienna) Vertex cuts / 27
52 Generalization of Stallings Structure Theorem Theorem (Stallings structure theorem, 1971) Let G be a finitely generated group. Then G has more than one end G splits over a finite subgroup. Bernhard Krön (Uni Vienna) Vertex cuts / 27
53 Generalization of Stallings Structure Theorem Theorem (Stallings structure theorem, 1971) Let G be a finitely generated group. Then G has more than one end G splits over a finite subgroup. Theorem (Dunwoody, Krön) Let G be any group. Then G has a Cayley-graph with more than one end G splits over a finite subgroup. Bernhard Krön (Uni Vienna) Vertex cuts / 27
54 Bernhard Krön (Uni Vienna) Vertex cuts / 27
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