On 2vertexconnected orientations of graphs


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1 On 2vertexconnected orientations of graphs Zoltán Szigeti Laboratoire GSCOP INP Grenoble, France 12 January 2012 Joint work with : Joseph Cheriyan (Waterloo) and Olivier Durand de Gevigney (Grenoble) Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
2 Outline Definitions Orientations with arcconnectivity constraints Orientations with vertexconnectivity constraints kvertexconnected orientations 2vertexconnected orientations Packing of special spanning subgraphs Matroid theory Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
3 Orientation G Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
4 Orientation G Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected 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, kedgeconnected if D X is connected X E, X k 1, kvertexconnected 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, karcconnected if D X is strongly connected X A, X k 1, kvertexconn. if D X is strongly connected X V, X = k 1 and V > k. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
6 karcconnected orientation Theorem (NashWilliams 1960) Given an undirected graph G, there exists a karcconnected orientation of G G is 2kedgeconnected. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
7 karcconnected orientation Theorem (NashWilliams 1960) Given an undirected graph G, there exists a karcconnected orientation of G G is 2kedgeconnected. necessity : k k X V X G Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
8 karcconnected orientation Theorem (NashWilliams 1960) Given an undirected graph G, there exists a karcconnected orientation of G G is 2kedgeconnected. necessity : 2k X V X G Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
9 kvertexconnected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)vertexconnected graph has a kvertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
10 kvertexconnected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)vertexconnected graph has a kvertexconnected orientation. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > k, there exists a kvertexconnected orientation of G G X is (2k 2 X )edgeconnected for all X V with X < k. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
11 kvertexconnected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)vertexconnected graph has a kvertexconnected orientation. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > k, there exists a kvertexconnected orientation of G G X is (2k 2 X )edgeconnected for all X V with X < k. necessity : G is kvertexconnected Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
12 kvertexconnected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)vertexconnected graph has a kvertexconnected orientation. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > k, there exists a kvertexconnected orientation of G G X is (2k 2 X )edgeconnected for all X V with X < k. necessity : X G is kvertexconnected Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
13 kvertexconnected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)vertexconnected graph has a kvertexconnected orientation. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > k, there exists a kvertexconnected orientation of G G X is (2k 2 X )edgeconnected for all X V with X < k. necessity : G X is (k X )vertexconnected Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
14 kvertexconnected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)vertexconnected graph has a kvertexconnected orientation. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > k, there exists a kvertexconnected orientation of G G X is (2k 2 X )edgeconnected for all X V with X < k. necessity : G X is (2k 2 X )edgeconnected Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
15 kvertexconnected orientation Conjecture (Thomassen 1989) There exists a function f (k) such that every f (k)vertexconnected graph has a kvertexconnected orientation. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > k, Remark there exists a kvertexconnected orientation of G G X is (2k 2 X )edgeconnected for all X V with X < k. Frank s conjecture would imply that f (k) 2k. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
16 2vertexconnected orientation : Frank s Conjecture Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > 2, there exists a 2vertexconnected orientation of G G is 4edgeconnected and G v is 2edgeconnected for all v V. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
17 2vertexconnected orientation : Frank s Conjecture Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > 2, there exists a 2vertexconnected orientation of G G is 4edgeconnected and G v is 2edgeconnected for all v V. Theorem (BergJordán 2006) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2vertexconnected Eulerian orientation of G G is 4edgeconnected and G v is 2edgeconnected for all v V. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
18 2vertexconnected orientation : Thomassen s Conjecture Theorem (Jordán 2005) Every 18vertexconnected graph has a 2vertexconnected orientation : f (2) 18. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
19 2vertexconnected orientation : Thomassen s Conjecture Theorem (Jordán 2005) Every 18vertexconnected graph has a 2vertexconnected orientation : f (2) 18. Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every 14vertexconnected graph has a 2vertexconnected orientation : f (2) 14. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
20 2vertexconnected orientation : Thomassen s Conjecture Theorem (Jordán 2005) Every 18vertexconnected graph has a 2vertexconnected orientation : f (2) 18. Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every 14vertexconnected graph has a 2vertexconnected orientation : f (2) 14. Remark Frank s conjecture would imply that f (2) 4. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
21 Proof Theorem (Jordán 2005) Every 18vertexconnected graph has a 2vertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
22 Proof Theorem (Jordán 2005) Every 18vertexconnected graph has a 2vertexconnected orientation. Theorem (BergJordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2vertexconnected orientation of G G v is 2edgeconnected for all v V. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
23 Proof Theorem (Jordán 2005) Every 18vertexconnected graph has a 2vertexconnected orientation. Theorem (BergJordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2vertexconnected orientation of G G v is 2edgeconnected for all v V. Find a spanning subgraph G such that G v is 2edgeconnected for all v V, = G v contains 2 edgedisjoint connected spanning subgraphs for all v V, = G contains 2 edgedisjoint 2vertexconnected spanning subgraphs. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
24 Proof Theorem (Jordán 2005) Every 18vertexconnected graph has a 2vertexconnected orientation. Theorem (BergJordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2vertexconnected orientation of G G v is 2edgeconnected for all v V. Find a spanning subgraph G such that G v is 2edgeconnected for all v V, = G v contains 2 edgedisjoint connected spanning subgraphs for all v V, = G contains 2 edgedisjoint 2vertexconnected spanning subgraphs. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
25 Proof Theorem (Jordán 2005) Every 18vertexconnected graph has a 2vertexconnected orientation. Theorem (BergJordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2vertexconnected orientation of G G v is 2edgeconnected for all v V. Find a spanning subgraph G such that G v is 2edgeconnected for all v V, = G v contains 2 edgedisjoint connected spanning subgraphs for all v V, = G contains 2 edgedisjoint 2vertexconnected spanning subgraphs. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
26 Proof Theorem (Jordán 2005) Every 18vertexconnected graph has a 2vertexconnected orientation. Theorem (BergJordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2vertexconnected orientation of G G v is 2edgeconnected for all v V. Theorem (Jordán 2005) (Tool 2) Every 6kvertexconnected graph contains k edgedisjoint 2vertexconnected spanning subgraphs. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
27 Proof (Jordán 2005) Let H be a 18vertexconnected graph. By Tool 2, H contains 3 edgedisjoint 2vertexconnected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2edgeconnected for every vertex v. Let T be the set of odd degree vertices in G. By Tool 1, G, and hence H, has a 2vertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
28 Proof (Jordán 2005) Let H be a 18vertexconnected graph. By Tool 2, H contains 3 edgedisjoint 2vertexconnected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2edgeconnected for every vertex v. Let T be the set of odd degree vertices in G. By Tool 1, G, and hence H, has a 2vertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
29 Proof (Jordán 2005) Let H be a 18vertexconnected graph. By Tool 2, H contains 3 edgedisjoint 2vertexconnected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2edgeconnected for every vertex v. Let T be the set of odd degree vertices in G. By Tool 1, G, and hence H, has a 2vertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
30 Proof (Jordán 2005) Let H be a 18vertexconnected graph. By Tool 2, H contains 3 edgedisjoint 2vertexconnected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2edgeconnected for every vertex v. Let T be the set of odd degree vertices in G. By Tool 1, G, and hence H, has a 2vertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
31 Proof (Jordán 2005) Let H be a 18vertexconnected graph. By Tool 2, H contains 3 edgedisjoint 2vertexconnected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2edgeconnected for every vertex v. Let T be the set of odd degree vertices in G. Since G 3 is connected, it contains a Tjoin F. By Tool 1, G, and hence H, has a 2vertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
32 Proof (Jordán 2005) Let H be a 18vertexconnected graph. By Tool 2, H contains 3 edgedisjoint 2vertexconnected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2edgeconnected for every vertex v. Let T be the set of odd degree vertices in G. Since G 3 is connected, it contains a Tjoin F. Then G:= G + F is Eulerian. By Tool 1, G, and hence H, has a 2vertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
33 Proof (Jordán 2005) Let H be a 18vertexconnected graph. By Tool 2, H contains 3 edgedisjoint 2vertexconnected spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2edgeconnected for every vertex v. Let T be the set of odd degree vertices in G. Since G 3 is connected, it contains a Tjoin F. Then G:= G + F is Eulerian. By Tool 1, G, and hence H, has a 2vertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
34 Proof of the new upperbound Theorem Every 14vertexconnected graph has a 2vertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
35 Proof of the new upperbound Theorem Every 14vertexconnected graph has a 2vertexconnected orientation. Theorem (BergJordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2vertexconnected orientation of G G v is 2edgeconnected for all v V. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
36 Proof of the new upperbound Theorem Every 14vertexconnected graph has a 2vertexconnected orientation. Theorem (BergJordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2vertexconnected orientation of G G v is 2edgeconnected for all v V. Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) (Tool 2 ) Every (6k + 2l)vertexconnected graph contains k 2vertexconnected and l connected edgedisjoint spanning subgraphs. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
37 Proof of the new upperbound Let H be a 14vertexconnected graph. By Tool 2, H contains two 2vertexconnected and one connected edgedisjoint spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2edgeconnected for every vertex v. Let T be the set of odd degree vertices in G. By Tool 1, G, and hence H, has a 2vertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
38 Proof of the new upperbound Let H be a 14vertexconnected graph. By Tool 2, H contains two 2vertexconnected and one connected edgedisjoint spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2edgeconnected for every vertex v. Let T be the set of odd degree vertices in G. By Tool 1, G, and hence H, has a 2vertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
39 Proof of the new upperbound Let H be a 14vertexconnected graph. By Tool 2, H contains two 2vertexconnected and one connected edgedisjoint spanning subgraphs : G 1,G 2, and G 3. Let G := G 1 G 2. Then G v is 2edgeconnected for every vertex v. Let T be the set of odd degree vertices in G. Since G 3 is connected, it contains a Tjoin F. Then G:= G + F is Eulerian. By Tool 1, G, and hence H, has a 2vertexconnected orientation. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
40 How to prove them? Theorem (BergJordán 2006) (Tool 1) Given an Eulerian graph G = (V,E) with V > 2, there exists a 2vertexconnected orientation of G G v is 2edgeconnected for all v V. Theorem (Jordán 2005) (Tool 2) Every 6kvertexconnected graph contains k edgedisjoint 2vertexconnected spanning subgraphs. Theorem (Tool 2 ) Every (6k + 2l)vertexconnected graph contains k 2vertexconnected and l connected edgedisjoint spanning subgraphs. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected 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 } (LovászYemini 1982). G is rigid if r R (E) = 2n 3 (Laman 1970). i(x) = 3 X Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected 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 } (LovászYemini 1982). G is rigid if r R (E) = 2n 3 (Laman 1970). a covering H of E Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected 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 } (LovászYemini 1982). G is rigid if r R (E) = 2n 3 (Laman 1970). a rigid graph Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
44 Packing of rigid spanning subgraphs Remark Every rigid graph is 2vertexconnected. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
45 Packing of rigid spanning subgraphs Remark Every rigid graph is 2vertexconnected. Proof Suppose G is not 2vertexconnected. 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 (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
46 Packing of rigid spanning subgraphs Remark Every rigid graph is 2vertexconnected. Proof Suppose G is not 2vertexconnected. 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 (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
47 Packing of rigid spanning subgraphs Remark Every rigid graph is 2vertexconnected. Proof Suppose G is not 2vertexconnected. 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 (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
48 Packing of rigid spanning subgraphs Remark Every rigid graph is 2vertexconnected. Proof Suppose G is not 2vertexconnected. 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 (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
49 Packing of rigid spanning subgraphs Remark Every rigid graph is 2vertexconnected. Theorem (LovászYemini 1982) Every 6vertexconnected graph is rigid. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
50 Packing of rigid spanning subgraphs Remark Every rigid graph is 2vertexconnected. Theorem (LovászYemini 1982) Every 6vertexconnected graph is rigid. Theorem (Jordán 2005) Every 6kvertexconnected graph contains k rigid edgedisjoint spanning subgraphs. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected 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 (GSCOP, Grenoble) On 2vertexconnected 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 edgedisjoint spanning trees of G for every partition P of V, G, l = 2 Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected 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 edgedisjoint spanning trees of G for every partition P of V, F 1 F 2 Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected 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 edgedisjoint spanning trees of G for every partition P of V, P Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected 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 edgedisjoint spanning trees of G for every partition P of V, E(P) E(P) Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected 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 edgedisjoint spanning trees of G for every partition P of V, E(P) l( P 1). F 1 F 2 E(P) Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected 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 edgedisjoint spanning trees of G for every partition P of V, E(P) l( P 1). Remark Every 2ledgeconnected graph contains l edgedisjoint spanning trees. E(P) = 1 2 P P d(p) 1 22l P > l( P 1). Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
58 Packing of rigid and connected spanning subgraphs Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every (6k + 2l)vertexconnected graph contains k rigid and l connected edgedisjoint spanning subgraphs. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
59 Packing of rigid and connected spanning subgraphs Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every (6k + 2l)vertexconnected graph contains k rigid and l connected edgedisjoint 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 edgedisjoint spanning subgraphs r Mk,l (E) = k(2n 3) + l(n 1). Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
60 Packing of rigid and connected spanning subgraphs Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every (6k + 2l)vertexconnected graph contains k rigid and l connected edgedisjoint 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 edgedisjoint spanning subgraphs r Mk,l (E) = k(2n 3) + l(n 1). Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
61 Packing of rigid and connected spanning subgraphs Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every (6k + 2l)vertexconnected graph contains k rigid and l connected edgedisjoint 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 edgedisjoint spanning subgraphs r Mk,l (E) = k(2n 3) + l(n 1). Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
62 Packing of rigid and connected spanning subgraphs Theorem (Cheriyan, Durand de Gevigney, Szigeti 2011) Every (6k + 2l)vertexconnected graph contains k rigid and l connected edgedisjoint 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 edgedisjoint spanning subgraphs r Mk,l (E) = k(2n 3) + l(n 1). Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
63 Remarks Remarks Proof of Jordán 2005 follows the proof of LovászYemini Our proof is completely different. It provides a transparent proof for the theorem of LovászYemini It enabled us to weaken the condition : instead of (6k + 2l)vertexconnectivity we used (6k + 2l,2k)connectivity. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
64 Remarks Remarks Proof of Jordán 2005 follows the proof of LovászYemini Our proof is completely different. It provides a transparent proof for the theorem of LovászYemini It enabled us to weaken the condition : instead of (6k + 2l)vertexconnectivity we used (6k + 2l,2k)connectivity. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
65 Remarks Remarks Proof of Jordán 2005 follows the proof of LovászYemini Our proof is completely different. It provides a transparent proof for the theorem of LovászYemini It enabled us to weaken the condition : instead of (6k + 2l)vertexconnectivity we used (6k + 2l,2k)connectivity. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
66 Remarks Remarks Proof of Jordán 2005 follows the proof of LovászYemini Our proof is completely different. It provides a transparent proof for the theorem of LovászYemini It enabled us to weaken the condition : instead of (6k + 2l)vertexconnectivity we used (6k + 2l,2k)connectivity. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
67 Remarks Remarks Proof of Jordán 2005 follows the proof of LovászYemini Our proof is completely different. It provides a transparent proof for the theorem of LovászYemini 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 )edgeconn. X V. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
68 Remarks Remarks Proof of Jordán 2005 follows the proof of LovászYemini Our proof is completely different. It provides a transparent proof for the theorem of LovászYemini 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 )edgeconn. X V. Conjecture (Frank 1995) Given an undirected graph G = (V,E) with V > 2, there exists a 2vertexconnected orientation of G G is (4, 2)connected. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
69 Main result Theorem (LovászYemini 1982) Every 6vertexconnected graph is rigid. Theorem (Jordán 2005) Every 6kvertexconnected graph contains k rigid edgedisjoint 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 (2vertexconnected) and l connected edgedisjoint spanning subgraphs. Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
70 Thank you for your attention! Z. Szigeti (GSCOP, Grenoble) On 2vertexconnected orientations 12 January / 21
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