# On 2-vertex-connected orientations of graphs

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1 On 2-vertex-connected orientations of graphs Zoltán Szigeti Laboratoire G-SCOP INP Grenoble, France 12 January 2012 Joint work with : Joseph Cheriyan (Waterloo) and Olivier Durand de Gevigney (Grenoble) Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

2 Outline Definitions Orientations with arc-connectivity constraints Orientations with vertex-connectivity constraints k-vertex-connected orientations 2-vertex-connected orientations Packing of special spanning subgraphs Matroid theory Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

3 Orientation G Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

4 Orientation G Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

5 Connectivity Definitions An undirected graph G = (V,E) is connected if there exists a (u, v)-path u, v V, k-edge-connected if D X is connected X E, X k 1, k-vertex-connected if D X is connected X V, X = k 1 and V > k. A directed graph D = (V,A) is strongly connected if a directed (u,v)-path (u,v) V V, k-arc-connected if D X is strongly connected X A, X k 1, k-vertex-conn. if D X is strongly connected X V, X = k 1 and V > k. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

6 k-arc-connected orientation Theorem (Nash-Williams 1960) Given an undirected graph G, there exists a k-arc-connected orientation of G G is 2k-edge-connected. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

7 k-arc-connected orientation Theorem (Nash-Williams 1960) Given an undirected graph G, there exists a k-arc-connected orientation of G G is 2k-edge-connected. necessity : k k X V X G Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

8 k-arc-connected orientation Theorem (Nash-Williams 1960) Given an undirected graph G, there exists a k-arc-connected orientation of G G is 2k-edge-connected. necessity : 2k X V X G Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

9 k-vertex-connected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)-vertex-connected graph has a k-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

10 k-vertex-connected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)-vertex-connected graph has a k-vertex-connected orientation. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > k, there exists a k-vertex-connected orientation of G G X is (2k 2 X )-edge-connected for all X V with X < k. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

11 k-vertex-connected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)-vertex-connected graph has a k-vertex-connected orientation. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > k, there exists a k-vertex-connected orientation of G G X is (2k 2 X )-edge-connected for all X V with X < k. necessity : G is k-vertex-connected Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

12 k-vertex-connected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)-vertex-connected graph has a k-vertex-connected orientation. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > k, there exists a k-vertex-connected orientation of G G X is (2k 2 X )-edge-connected for all X V with X < k. necessity : X G is k-vertex-connected Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

13 k-vertex-connected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)-vertex-connected graph has a k-vertex-connected orientation. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > k, there exists a k-vertex-connected orientation of G G X is (2k 2 X )-edge-connected for all X V with X < k. necessity : G X is (k X )-vertex-connected Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

14 k-vertex-connected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)-vertex-connected graph has a k-vertex-connected orientation. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > k, there exists a k-vertex-connected orientation of G G X is (2k 2 X )-edge-connected for all X V with X < k. necessity : G X is (2k 2 X )-edge-connected Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

15 k-vertex-connected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)-vertex-connected graph has a k-vertex-connected orientation. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > k, Remark there exists a k-vertex-connected orientation of G G X is (2k 2 X )-edge-connected for all X V with X < k. Frank s conjecture would imply that f (k) 2k. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

16 2-vertex-connected orientation : Frank s Conjecture Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > 2, there exists a 2-vertex-connected orientation of G G is 4-edge-connected and G v is 2-edge-connected for all v V. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

17 2-vertex-connected orientation : Frank s Conjecture Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > 2, there exists a 2-vertex-connected orientation of G G is 4-edge-connected and G v is 2-edge-connected for all v V. Theorem (Berg-Jordán 2006) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2-vertex-connected Eulerian orientation of G G is 4-edge-connected and G v is 2-edge-connected for all v V. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

18 2-vertex-connected orientation : Thomassen s Conjecture Theorem (Jordán 2005) Every 18-vertex-connected graph has a 2-vertex-connected orientation : f (2) 18. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

19 2-vertex-connected orientation : Thomassen s Conjecture Theorem (Jordán 2005) Every 18-vertex-connected graph has a 2-vertex-connected orientation : f (2) 18. Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every 14-vertex-connected graph has a 2-vertex-connected orientation : f (2) 14. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

20 2-vertex-connected orientation : Thomassen s Conjecture Theorem (Jordán 2005) Every 18-vertex-connected graph has a 2-vertex-connected orientation : f (2) 18. Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every 14-vertex-connected graph has a 2-vertex-connected orientation : f (2) 14. Remark Frank s conjecture would imply that f (2) 4. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

21 Proof Theorem (Jordán 2005) Every 18-vertex-connected graph has a 2-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

22 Proof Theorem (Jordán 2005) Every 18-vertex-connected graph has a 2-vertex-connected orientation. Theorem (Berg-Jordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2-vertex-connected orientation of G G v is 2-edge-connected for all v V. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

23 Proof Theorem (Jordán 2005) Every 18-vertex-connected graph has a 2-vertex-connected orientation. Theorem (Berg-Jordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2-vertex-connected orientation of G G v is 2-edge-connected for all v V. Find a spanning subgraph G such that G v is 2-edge-connected for all v V, = G v contains 2 edge-disjoint connected spanning subgraphs for all v V, = G contains 2 edge-disjoint 2-vertex-connected spanning subgraphs. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

24 Proof Theorem (Jordán 2005) Every 18-vertex-connected graph has a 2-vertex-connected orientation. Theorem (Berg-Jordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2-vertex-connected orientation of G G v is 2-edge-connected for all v V. Find a spanning subgraph G such that G v is 2-edge-connected for all v V, = G v contains 2 edge-disjoint connected spanning subgraphs for all v V, = G contains 2 edge-disjoint 2-vertex-connected spanning subgraphs. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

25 Proof Theorem (Jordán 2005) Every 18-vertex-connected graph has a 2-vertex-connected orientation. Theorem (Berg-Jordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2-vertex-connected orientation of G G v is 2-edge-connected for all v V. Find a spanning subgraph G such that G v is 2-edge-connected for all v V, = G v contains 2 edge-disjoint connected spanning subgraphs for all v V, = G contains 2 edge-disjoint 2-vertex-connected spanning subgraphs. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

26 Proof Theorem (Jordán 2005) Every 18-vertex-connected graph has a 2-vertex-connected orientation. Theorem (Berg-Jordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2-vertex-connected orientation of G G v is 2-edge-connected for all v V. Theorem (Jordán 2005) (Tool 2) Every 6k-vertex-connected graph contains k edge-disjoint 2-vertexconnected spanning subgraphs. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

27 Proof (Jordán 2005) Let H be a 18-vertex-connected graph. By Tool 2, H contains 3 edge-disjoint 2-vertex-connected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2-edge-connected for every vertex v. Let T be the set of odd degree vertices in G. By Tool 1, G, and hence H, has a 2-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

28 Proof (Jordán 2005) Let H be a 18-vertex-connected graph. By Tool 2, H contains 3 edge-disjoint 2-vertex-connected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2-edge-connected for every vertex v. Let T be the set of odd degree vertices in G. By Tool 1, G, and hence H, has a 2-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

29 Proof (Jordán 2005) Let H be a 18-vertex-connected graph. By Tool 2, H contains 3 edge-disjoint 2-vertex-connected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2-edge-connected for every vertex v. Let T be the set of odd degree vertices in G. By Tool 1, G, and hence H, has a 2-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

30 Proof (Jordán 2005) Let H be a 18-vertex-connected graph. By Tool 2, H contains 3 edge-disjoint 2-vertex-connected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2-edge-connected for every vertex v. Let T be the set of odd degree vertices in G. By Tool 1, G, and hence H, has a 2-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

31 Proof (Jordán 2005) Let H be a 18-vertex-connected graph. By Tool 2, H contains 3 edge-disjoint 2-vertex-connected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2-edge-connected for every vertex v. Let T be the set of odd degree vertices in G. Since G 3 is connected, it contains a T-join F. By Tool 1, G, and hence H, has a 2-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

32 Proof (Jordán 2005) Let H be a 18-vertex-connected graph. By Tool 2, H contains 3 edge-disjoint 2-vertex-connected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2-edge-connected for every vertex v. Let T be the set of odd degree vertices in G. Since G 3 is connected, it contains a T-join F. Then G:= G + F is Eulerian. By Tool 1, G, and hence H, has a 2-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

33 Proof (Jordán 2005) Let H be a 18-vertex-connected graph. By Tool 2, H contains 3 edge-disjoint 2-vertex-connected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2-edge-connected for every vertex v. Let T be the set of odd degree vertices in G. Since G 3 is connected, it contains a T-join F. Then G:= G + F is Eulerian. By Tool 1, G, and hence H, has a 2-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

34 Proof of the new upperbound Theorem Every 14-vertex-connected graph has a 2-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

35 Proof of the new upperbound Theorem Every 14-vertex-connected graph has a 2-vertex-connected orientation. Theorem (Berg-Jordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2-vertex-connected orientation of G G v is 2-edge-connected for all v V. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

36 Proof of the new upperbound Theorem Every 14-vertex-connected graph has a 2-vertex-connected orientation. Theorem (Berg-Jordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2-vertex-connected orientation of G G v is 2-edge-connected for all v V. Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) (Tool 2 ) Every (6k + 2l)-vertex-connected graph contains k 2-vertex-connected and l connected edge-disjoint spanning subgraphs. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

37 Proof of the new upperbound Let H be a 14-vertex-connected graph. By Tool 2, H contains two 2-vertex-connected and one connected edge-disjoint spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2-edge-connected for every vertex v. Let T be the set of odd degree vertices in G. By Tool 1, G, and hence H, has a 2-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

38 Proof of the new upperbound Let H be a 14-vertex-connected graph. By Tool 2, H contains two 2-vertex-connected and one connected edge-disjoint spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2-edge-connected for every vertex v. Let T be the set of odd degree vertices in G. By Tool 1, G, and hence H, has a 2-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

39 Proof of the new upperbound Let H be a 14-vertex-connected graph. By Tool 2, H contains two 2-vertex-connected and one connected edge-disjoint spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2-edge-connected for every vertex v. Let T be the set of odd degree vertices in G. Since G 3 is connected, it contains a T-join F. Then G:= G + F is Eulerian. By Tool 1, G, and hence H, has a 2-vertex-connected orientation. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

40 How to prove them? Theorem (Berg-Jordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2-vertex-connected orientation of G G v is 2-edge-connected for all v V. Theorem (Jordán 2005) (Tool 2) Every 6k-vertex-connected graph contains k edge-disjoint 2-vertexconnected spanning subgraphs. Theorem (Tool 2 ) Every (6k + 2l)-vertex-connected graph contains k 2-vertex-connected and l connected edge-disjoint spanning subgraphs. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

41 Rigidity Matroid Definition Given a graph G= (V,E) with n= V. Rigidity Matroid : independent sets : R(G)= {F E : i F (X) 2 X 3 X V } (Crapo 1979). rank function r R (F)= min{ X H (2 X 3) : H set of subsets of V covering F } (Lovász-Yemini 1982). G is rigid if r R (E) = 2n 3 (Laman 1970). i(x) = 3 X Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

42 Rigidity Matroid Definition Given a graph G= (V,E) with n= V. Rigidity Matroid : independent sets : R(G)= {F E : i F (X) 2 X 3 X V } (Crapo 1979). rank function r R (F)= min{ X H (2 X 3) : H set of subsets of V covering F } (Lovász-Yemini 1982). G is rigid if r R (E) = 2n 3 (Laman 1970). a covering H of E Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

43 Rigidity Matroid Definition Given a graph G= (V,E) with n= V. Rigidity Matroid : independent sets : R(G)= {F E : i F (X) 2 X 3 X V } (Crapo 1979). rank function r R (F)= min{ X H (2 X 3) : H set of subsets of V covering F } (Lovász-Yemini 1982). G is rigid if r R (E) = 2n 3 (Laman 1970). a rigid graph Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

44 Packing of rigid spanning subgraphs Remark Every rigid graph is 2-vertex-connected. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

45 Packing of rigid spanning subgraphs Remark Every rigid graph is 2-vertex-connected. Proof Suppose G is not 2-vertex-connected. Then there exists a covering {X,Y } of E such that X Y 1. r R (E) 2 X Y 3 = 2 X Y + 2 X Y 6 2n 4. G is not rigid. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

46 Packing of rigid spanning subgraphs Remark Every rigid graph is 2-vertex-connected. Proof Suppose G is not 2-vertex-connected. Then there exists a covering {X,Y } of E such that X Y 1. r R (E) 2 X Y 3 = 2 X Y + 2 X Y 6 2n 4. G is not rigid. X Y Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

47 Packing of rigid spanning subgraphs Remark Every rigid graph is 2-vertex-connected. Proof Suppose G is not 2-vertex-connected. Then there exists a covering {X,Y } of E such that X Y 1. r R (E) 2 X Y 3 = 2 X Y + 2 X Y 6 2n 4. G is not rigid. X Y Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

48 Packing of rigid spanning subgraphs Remark Every rigid graph is 2-vertex-connected. Proof Suppose G is not 2-vertex-connected. Then there exists a covering {X,Y } of E such that X Y 1. r R (E) 2 X Y 3 = 2 X Y + 2 X Y 6 2n 4. G is not rigid. X Y Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

49 Packing of rigid spanning subgraphs Remark Every rigid graph is 2-vertex-connected. Theorem (Lovász-Yemini 1982) Every 6-vertex-connected graph is rigid. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

50 Packing of rigid spanning subgraphs Remark Every rigid graph is 2-vertex-connected. Theorem (Lovász-Yemini 1982) Every 6-vertex-connected graph is rigid. Theorem (Jordán 2005) Every 6k-vertex-connected graph contains k rigid edge-disjoint spanning subgraphs. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

51 Circuit Matroid Definition Given a graph G= (V,E) with n= V. Circuit Matroid : independent sets : C(G) = the edge sets of the forests of G. rank function r C (F)= n c(f). G is connected if a spanning tree (r C (E) = n 1). Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

52 Packing of connected spanning subgraphs Theorem (Tutte 1961) Given an undirected graph G and an integer l 1, there exist l edge-disjoint spanning trees of G for every partition P of V, G, l = 2 Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

53 Packing of connected spanning subgraphs Theorem (Tutte 1961) Given an undirected graph G and an integer l 1, there exist l edge-disjoint spanning trees of G for every partition P of V, F 1 F 2 Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

54 Packing of connected spanning subgraphs Theorem (Tutte 1961) Given an undirected graph G and an integer l 1, there exist l edge-disjoint spanning trees of G for every partition P of V, P Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

55 Packing of connected spanning subgraphs Theorem (Tutte 1961) Given an undirected graph G and an integer l 1, there exist l edge-disjoint spanning trees of G for every partition P of V, E(P) E(P) Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

56 Packing of connected spanning subgraphs Theorem (Tutte 1961) Given an undirected graph G and an integer l 1, there exist l edge-disjoint spanning trees of G for every partition P of V, E(P) l( P 1). F 1 F 2 E(P) Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

57 Packing of connected spanning subgraphs Theorem (Tutte 1961) Given an undirected graph G and an integer l 1, there exist l edge-disjoint spanning trees of G for every partition P of V, E(P) l( P 1). Remark Every 2l-edge-connected graph contains l edge-disjoint spanning trees. E(P) = 1 2 P P d(p) 1 22l P > l( P 1). Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

58 Packing of rigid and connected spanning subgraphs Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every (6k + 2l)-vertex-connected graph contains k rigid and l connected edge-disjoint spanning subgraphs. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

59 Packing of rigid and connected spanning subgraphs Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every (6k + 2l)-vertex-connected graph contains k rigid and l connected edge-disjoint spanning subgraphs. Tool M k,l (G) = matroid union of k copies of R(G) and l copies of C(G). independent sets are the union of k independent sets of R(G) and l independent sets of C(G). rank r Mk,l (E)= min F E kr R (F) + lr C (F) + E \ F. (Edmonds 1968) G contains k rigid and l connected edge-disjoint spanning subgraphs r Mk,l (E) = k(2n 3) + l(n 1). Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

60 Packing of rigid and connected spanning subgraphs Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every (6k + 2l)-vertex-connected graph contains k rigid and l connected edge-disjoint spanning subgraphs. Tool M k,l (G) = matroid union of k copies of R(G) and l copies of C(G). independent sets are the union of k independent sets of R(G) and l independent sets of C(G). rank r Mk,l (E)= min F E kr R (F) + lr C (F) + E \ F. (Edmonds 1968) G contains k rigid and l connected edge-disjoint spanning subgraphs r Mk,l (E) = k(2n 3) + l(n 1). Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

61 Packing of rigid and connected spanning subgraphs Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every (6k + 2l)-vertex-connected graph contains k rigid and l connected edge-disjoint spanning subgraphs. Tool M k,l (G) = matroid union of k copies of R(G) and l copies of C(G). independent sets are the union of k independent sets of R(G) and l independent sets of C(G). rank r Mk,l (E)= min F E kr R (F) + lr C (F) + E \ F. (Edmonds 1968) G contains k rigid and l connected edge-disjoint spanning subgraphs r Mk,l (E) = k(2n 3) + l(n 1). Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

62 Packing of rigid and connected spanning subgraphs Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every (6k + 2l)-vertex-connected graph contains k rigid and l connected edge-disjoint spanning subgraphs. Tool M k,l (G) = matroid union of k copies of R(G) and l copies of C(G). independent sets are the union of k independent sets of R(G) and l independent sets of C(G). rank r Mk,l (E)= min F E kr R (F) + lr C (F) + E \ F. (Edmonds 1968) G contains k rigid and l connected edge-disjoint spanning subgraphs r Mk,l (E) = k(2n 3) + l(n 1). Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

63 Remarks Remarks Proof of Jordán 2005 follows the proof of Lovász-Yemini Our proof is completely different. It provides a transparent proof for the theorem of Lovász-Yemini It enabled us to weaken the condition : instead of (6k + 2l)-vertexconnectivity we used (6k + 2l,2k)-connectivity. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

64 Remarks Remarks Proof of Jordán 2005 follows the proof of Lovász-Yemini Our proof is completely different. It provides a transparent proof for the theorem of Lovász-Yemini It enabled us to weaken the condition : instead of (6k + 2l)-vertexconnectivity we used (6k + 2l,2k)-connectivity. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

65 Remarks Remarks Proof of Jordán 2005 follows the proof of Lovász-Yemini Our proof is completely different. It provides a transparent proof for the theorem of Lovász-Yemini It enabled us to weaken the condition : instead of (6k + 2l)-vertexconnectivity we used (6k + 2l,2k)-connectivity. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

66 Remarks Remarks Proof of Jordán 2005 follows the proof of Lovász-Yemini Our proof is completely different. It provides a transparent proof for the theorem of Lovász-Yemini It enabled us to weaken the condition : instead of (6k + 2l)-vertexconnectivity we used (6k + 2l,2k)-connectivity. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

67 Remarks Remarks Proof of Jordán 2005 follows the proof of Lovász-Yemini Our proof is completely different. It provides a transparent proof for the theorem of Lovász-Yemini It enabled us to weaken the condition : instead of (6k + 2l)-vertexconnectivity we used (6k + 2l,2k)-connectivity. Definition G is (a,b)-connected if G X is (a b X )-edge-conn. X V. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

68 Remarks Remarks Proof of Jordán 2005 follows the proof of Lovász-Yemini Our proof is completely different. It provides a transparent proof for the theorem of Lovász-Yemini It enabled us to weaken the condition : instead of (6k + 2l)-vertexconnectivity we used (6k + 2l,2k)-connectivity. Definition G is (a,b)-connected if G X is (a b X )-edge-conn. X V. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > 2, there exists a 2-vertex-connected orientation of G G is (4, 2)-connected. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

69 Main result Theorem (Lovász-Yemini 1982) Every 6-vertex-connected graph is rigid. Theorem (Jordán 2005) Every 6k-vertex-connected graph contains k rigid edge-disjoint spanning subgraphs. Theorem (Jackson et Jordán 2009) Every simple (6,2)-connected graph is rigid. Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every simple (6k + 2l, 2k)-connected graph contains k ( 1) rigid (2-vertex-connected) and l connected edge-disjoint spanning subgraphs. Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

70 Thank you for your attention! Z. Szigeti (G-SCOP, Grenoble) On 2-vertex-connected orientations 12 January / 21

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