Hamiltonian thickness and fault-tolerant spanning rooted path - - PowerPoint PPT Presentation

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Hamiltonian thickness and fault-tolerant spanning rooted path - - PowerPoint PPT Presentation

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Hamiltonian thickness and fault-tolerant spanning rooted path systems of graphs Yinfeng Zhu ( 6 ) Shanghai Jiao Tong University Dec 8, 2015 Joint with Yaokun Wu n & Ziqing Xiang ( f ) 1 / 20 Spanning connection

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SLIDE 1

Hamiltonian thickness and fault-tolerant spanning rooted path systems of graphs

Yinfeng Zhu (6Û¸)

Shanghai Jiao Tong University

Dec 8, 2015

Joint with Yaokun Wu £Çˆn¤& Ziqing Xiang (•f—)

1 / 20

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SLIDE 2

Spanning connection pattern with fault-tolerance

1 2 3 4 5 6 7 8 9 Deleting ≤ 2 vertices always results in a Hamiltonian graph.

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SLIDE 3

Spanning connection pattern with fault-tolerance

1 2 3 4 5 6 7 8 9 Deleting ≤ 2 vertices always results in a Hamiltonian graph.

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SLIDE 4

Spanning connection pattern with fault-tolerance

1 2 3 4 5 6 7 8 9 Deleting ≤ 2 vertices always results in a Hamiltonian graph.

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SLIDE 5

Spanning connection pattern with fault-tolerance

1 2 3 4 5 6 7 8 9 Deleting ≤ 2 vertices always results in a Hamiltonian graph.

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SLIDE 6

Spanning connection pattern with fault-tolerance

1 2 3 4 5 6 7 8 9 Deleting ≤ 2 vertices always results in a Hamiltonian graph.

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SLIDE 7

Hamiltonian thickness

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SLIDE 8

Hamiltonian thickness

1 2 3 4 5 6 7 8 9

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Hamiltonian thickness

1 2 3 4 5 6 7 8 9 A 4-thick Hamiltonian vertex ordering.

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SLIDE 10

Hamiltonian thickness

1 2 3 4 5 6 7 8 9 A 4-thick Hamiltonian vertex ordering.

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SLIDE 11

Hamiltonian thickness

1 2 3 4 5 6 7 8 9 ≥ 4 A 4-thick Hamiltonian vertex ordering.

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SLIDE 12

Hamiltonian thickness

1 2 3 4 5 6 7 8 9 ≥ 4 A 4-thick Hamiltonian vertex ordering.

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SLIDE 13

Hamiltonian thickness

1 2 3 4 5 6 7 8 9 ≥ 4 ≥ 4 A 4-thick Hamiltonian vertex ordering.

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SLIDE 14

Hamiltonian thickness

1 2 3 4 5 6 7 8 9 ≥ 4 ≥ 4 A 4-thick Hamiltonian vertex ordering.

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SLIDE 15

Hamiltonian thickness

1 2 3 4 5 6 7 8 9 ≥ 4 ≥ 4 ≥ 4 A 4-thick Hamiltonian vertex ordering.

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SLIDE 16

Hamiltonian thickness

1 2 3 4 5 6 7 8 9 ≥ 4 ≥ 4 ≥ 4 A 4-thick Hamiltonian vertex ordering.

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SLIDE 17

Hamiltonian thickness

1 2 3 4 5 6 7 8 9 ≥ 4 ≥ 4 ≥ 4 ≥ 4 ≥ 4 ≥ 4 A 4-thick Hamiltonian vertex ordering.

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SLIDE 18

Hamiltonian thickness

1 2 3 4 5 6 7 8 9 ≥ 4 ≥ 4 ≥ 4 ≥ 4 ≥ 4 ≥ 4 A 4-thick Hamiltonian vertex ordering.

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SLIDE 19

Hamiltonian thickness

1 2 3 4 5 6 7 8 9 ≥ 4 ≥ 4 ≥ 4 ≥ 4 ≥ 4 ≥ 4 4-clique A 4-thick Hamiltonian vertex ordering.

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SLIDE 20

Hamiltonian thickness

1 2 3 4 5 6 7 8 9 ≥ 4 ≥ 4 ≥ 4 ≥ 4 ≥ 4 ≥ 4 4-clique A 4-thick Hamiltonian vertex ordering. Definition A graph G is a k-thick Hamiltonian graph if G has a k-thick Hamiltonian vertex ordering.

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SLIDE 21

Path system I

Let G be a graph and let H be a multigraph with VH ⊆ VG and |EH| = m.

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SLIDE 22

Path system I

Let G be a graph and let H be a multigraph with VH ⊆ VG and |EH| = m. For each edge uv ∈ EH, an uv-path in G is a path in G whose endpoints are u and v.

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SLIDE 23

Path system I

Let G be a graph and let H be a multigraph with VH ⊆ VG and |EH| = m. For each edge uv ∈ EH, an uv-path in G is a path in G whose endpoints are u and v. A path in G with identical endpoints is either a trivial path (of length zero) or a cycle (of length at least three). For a loop edge vv ∈ EH, an vv-path in G should be understood as a cycle of length at least 3 but not any trivial path.

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SLIDE 24

Path system II

A path system of G rooted at EH is a set Q = {Pe : e ∈ EH} of m paths in G such that Pe is an e-path and every two distinct paths in the family intersect only at their possible common endpoints.

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SLIDE 25

Path system II

A path system of G rooted at EH is a set Q = {Pe : e ∈ EH} of m paths in G such that Pe is an e-path and every two distinct paths in the family intersect only at their possible common endpoints. A path system of G is spanning if the union of the vertices of the paths in it is the whole vertex set of G.

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SLIDE 26

Path system II

A path system of G rooted at EH is a set Q = {Pe : e ∈ EH} of m paths in G such that Pe is an e-path and every two distinct paths in the family intersect only at their possible common endpoints. A path system of G is spanning if the union of the vertices of the paths in it is the whole vertex set of G. s t s t H A spanning 3-rail rooted at EH

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SLIDE 27

Path system II

A path system of G rooted at EH is a set Q = {Pe : e ∈ EH} of m paths in G such that Pe is an e-path and every two distinct paths in the family intersect only at their possible common endpoints. A path system of G is spanning if the union of the vertices of the paths in it is the whole vertex set of G. s t s t H A spanning 3-rail rooted at EH spanning 1-rail Hamiltonian path spanning 2-rail Hamiltonian cycle

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SLIDE 28

f-factor

Let H be a multigraph and f : VH → N be a map. An f-factor of H is a multigraph F with VF = VH; EF ⊆ EH; degF(v) = f(v) for all v ∈ VF.(Each loop contributes 2 degrees.) The multigraph H is f-factor friendly if every g-factor of H can be extended into an f-factor of H by adding edges as long as g ≤ f.

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SLIDE 29

A path system of G rooted at (H, f)

Let G be a graph with VH ⊆ VG. A path system of G rooted at (H, f) is a path system of G − f −1(0) rooted at EF for some f-factor F of H. A path system of G rooted at (H, f) is called spanning if every vertex from VG \ f −1(0) appears in some path of the system. Note that f −1(0) can be thought of as the set of faulty nodes in G.

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SLIDE 30

Question and result

Assume that H is f-factor friendly.

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Question and result

Assume that H is f-factor friendly. Which kind of (sparse) graphs have a (spanning) path system rooted at (H, f)?

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Question and result

Assume that H is f-factor friendly. Which kind of (sparse) graphs have a (spanning) path system rooted at (H, f)? We show that G has a (spanning) path system rooted at (H, f) whenever the Hamiltonian thickness of G is no less than some parameter determined by f.

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SLIDE 33

Hamiltonian thickness vs. path system

Suppose that G has a vertex ordering π1, . . . , πn with Hamiltonian thickness at least k. Let H be an f-factor friendly multigraph.

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SLIDE 34

Hamiltonian thickness vs. path system

Suppose that G has a vertex ordering π1, . . . , πn with Hamiltonian thickness at least k. Let H be an f-factor friendly multigraph. The graph G has a path system rooted at (H, f) provided π1 ∈ VH and k is no less than (i) |f −1(0)| +

v∈VH f(v) −

1

2 minf(v)0 f(v)

  • + 1

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SLIDE 35

Hamiltonian thickness vs. path system

Suppose that G has a vertex ordering π1, . . . , πn with Hamiltonian thickness at least k. Let H be an f-factor friendly multigraph. The graph G has a path system rooted at (H, f) provided π1 ∈ VH and k is no less than (i) |f −1(0)| +

v∈VH f(v) −

1

2 minf(v)0 f(v)

  • + 1

(ii) |f −1(0)| +

v∈VH f(v) −

1

2 minf(v)0 f(v)

  • (H is loopless)

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SLIDE 36

Hamiltonian thickness vs. path system

Suppose that G has a vertex ordering π1, . . . , πn with Hamiltonian thickness at least k. Let H be an f-factor friendly multigraph. The graph G has a path system rooted at (H, f) provided π1 ∈ VH and k is no less than (i) |f −1(0)| +

v∈VH f(v) −

1

2 minf(v)0 f(v)

  • + 1

(ii) |f −1(0)| +

v∈VH f(v) −

1

2 minf(v)0 f(v)

  • (H is loopless)

(iii) |f −1(0)| +

v∈VH f(v) − minf(v)0 f(v)

(H is triangle-free and loopless)

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SLIDE 37

Hamiltonian thickness vs. path system

Suppose that G has a vertex ordering π1, . . . , πn with Hamiltonian thickness at least k. Let H be an f-factor friendly multigraph. The graph G has a path system rooted at (H, f) provided π1 ∈ VH and k is no less than (i) |f −1(0)| +

v∈VH f(v) −

1

2 minf(v)0 f(v)

  • + 1

(ii) |f −1(0)| +

v∈VH f(v) −

1

2 minf(v)0 f(v)

  • (H is loopless)

(iii) |f −1(0)| +

v∈VH f(v) − minf(v)0 f(v)

(H is triangle-free and loopless) (iv) |f −1(0)| + 1

2

  • v∈VH f(v)

(H is bipartite with mH(u, v) ≥ min{f(u), f(v)})

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SLIDE 38

Hamiltonian thickness vs. path system

Suppose that G has a vertex ordering π1, . . . , πn with Hamiltonian thickness at least k. Let H be an f-factor friendly multigraph. The graph G has a path system rooted at (H, f) provided π1 ∈ VH and k is no less than (i) |f −1(0)| +

v∈VH f(v) −

1

2 minf(v)0 f(v)

  • + 1

(ii) |f −1(0)| +

v∈VH f(v) −

1

2 minf(v)0 f(v)

  • (H is loopless)

(iii) |f −1(0)| +

v∈VH f(v) − minf(v)0 f(v)

(H is triangle-free and loopless) (iv) |f −1(0)| + 1

2

  • v∈VH f(v)

(H is bipartite with mH(u, v) ≥ min{f(u), f(v)})

If π1 VH, the bounds for thickness should be increased by one.

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SLIDE 39

Hamiltonian thickness vs. spanning path system

If πn ∈ VH and f(πn) ≥ 2, the same bound applies for the existence of a spanning path system. In remaining cases, we may need to increase the thickness bound by

  • ne.

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SLIDE 40

Fault-tolerant spanning k-rail

Corollary Let k and t be integers satisfying k ≥ t ≥ 2 and let G be a graph with a k-thick Hamiltonian vertex ordering π1, . . . , πn. For every C ⊆ VG\{π1,πn}

≤k−t

  • ,

G − C has a spanning t-rail between π1 and πn.

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SLIDE 41

Fault-tolerant spanning k-rail

Corollary Let k and t be integers satisfying k ≥ t ≥ 2 and let G be a graph with a k-thick Hamiltonian vertex ordering π1, . . . , πn. For every C ⊆ VG\{π1,πn}

≤k−t

  • ,

G − C has a spanning t-rail between π1 and πn. π1 πn t = 1, |C| = k − t, no spanning t-rail between π1 and πn in G − C.

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The greedy strategy, I

We start from a path family P which consists of f(v) trivial paths at vertex v for all v ∈ VH as well as one trivial path at v for all v ∈ VG \ VH. At each iteration, we find a suitable edge of G to connect two paths in P into one and update P accordingly. We always try to add the edge so that the path family obtained does not violate some obvious conditions for a path family to be further updated into a path system rooted at (H, f) (together with some other paths in G − VH, if that path system is not required to be spanning). A vertex v ∈ VG is finished in P if v ∈ VH and v is incident to degH(v) edges in P or v VH and v is incident to 2 edges in P. A vertex v ∈ VG is fresh in P if v VH and v is incident to no edges in P.

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The greedy strategy, II

Let π1, . . . , πn be an ordering of VG with high Hamiltonian thickness. (i) Find the leftmost unfinished vertex in P, say πi. If πi is fresh or πi ∈ VH, we choose P ∈ P as a trivial path at πi. Else, we choose P as the unique nontrivial path in P that contains πi as an endpoint. (ii) We try to find another unfinished vertex πj ∈ NG(πi) and a path Q ∈ P containing πj and connect P and Q by adding edge πiπj. We should guarantee that the updated path family is still “allowed”. (iii) If πi+1 is fresh, then we should choose j = i + 1.

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The greedy strategy, III

Suppose we want to find a spanning path system rooted at (H, f). We follow two more strategies. (i) Whenever possible, we should first try to choose πj so that the updated path family P does not contain a subdivision of any f-factor of H. (ii) Suppose πn ∈ VH and f(πn) ≥ 2. If πn is not on a nontrivial path in the path family P yet, then we choose j = n only when there is no

  • ther choice.

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Finding spanning path system

P = {0, 0, 1, 2, 4, 5, 6, 7, 7, 8, 9} Q = {} P =? Q =? f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3

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SLIDE 46

Finding spanning path system

P = {0, 0, 1, 2, 4, 5, 6, 7, 7, 8, 9} Q = {} P =0 Q =? f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3

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SLIDE 47

Finding spanning path system

P = {0, 0, 1, 2, 4, 5, 6, 7, 7, 8, 9} Q = {} P =0 Q =2 f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 2

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SLIDE 48

Finding spanning path system

P = {0, 0, 1, 2, 4, 5, 6, 7, 7, 8, 9} Q = {} P =0 Q =2 f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 2

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SLIDE 49

Finding spanning path system

P = {0-2, 0, 1, 4, 5, 6, 7, 7, 8, 9} Q = {} P =? Q =? f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3

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SLIDE 50

Finding spanning path system

P = {0-2, 0, 1, 4, 5, 6, 7, 7, 8, 9} Q = {} P =0 Q =? f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3

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SLIDE 51

Finding spanning path system

P = {0-2, 0, 1, 4, 5, 6, 7, 7, 8, 9} Q = {} P =0 Q =6 f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 6

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SLIDE 52

Finding spanning path system

P = {0-2, 0, 1, 4, 5, 6, 7, 7, 8, 9} Q = {} P =0 Q =6 f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 6

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SLIDE 53

Finding spanning path system

P = {0-2, 0-6, 1, 4, 5, 7, 7, 8, 9} Q = {} P =? Q =? f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3

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SLIDE 54

Finding spanning path system

P = {0-2, 0-6, 1, 4, 5, 7, 7, 8, 9} Q = {} P =1 Q =? f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 1

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SLIDE 55

Finding spanning path system

P = {0-2, 0-6, 1, 4, 5, 7, 7, 8, 9} Q = {} P =1 Q =5 f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 1 5

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SLIDE 56

Finding spanning path system

P = {0-2, 0-6, 1, 4, 5, 7, 7, 8, 9} Q = {} P =1 Q =5 f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 1 5

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SLIDE 57

Finding spanning path system

P = {0-2, 0-6, 1-5, 4, 7, 7, 8, 9} Q = {} P =? Q =? f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3

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SLIDE 58

Finding spanning path system

P = {0-2, 0-6, 1-5, 4, 7, 7, 8, 9} Q = {} P =0-2 Q =? f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 2

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SLIDE 59

Finding spanning path system

P = {0-2, 0-6, 1-5, 4, 7, 7, 8, 9} Q = {} P =0-2 Q =7 f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7

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SLIDE 60

Finding spanning path system

P = {0-2, 0-6, 1-5, 4, 7, 7, 8, 9} Q = {} P =0-2 Q =7 f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7

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SLIDE 61

Finding spanning path system

P = {0-6, 1-5, 4, 7, 8, 9} Q = {0-2-7} P =? Q =? f = 2 1 1 2 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7

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SLIDE 62

Finding spanning path system

P = {0-6, 1-5, 4, 7, 8, 9} Q = {0-2-7} P =4 Q =? f = 2 1 1 2 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 4

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SLIDE 63

Finding spanning path system

P = {0-6, 1-5, 4, 7, 8, 9} Q = {0-2-7} P =4 Q =8 f = 2 1 1 2 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 4 8

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SLIDE 64

Finding spanning path system

P = {0-6, 1-5, 4, 7, 8, 9} Q = {0-2-7} P =4 Q =8 f = 2 1 1 2 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 4 8

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SLIDE 65

Finding spanning path system

P = {0-6, 1-5, 4-8, 7, 9} Q = {0-2-7} P =? Q =? f = 2 1 1 2 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7

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SLIDE 66

Finding spanning path system

P = {0-6, 1-5, 4-8, 7, 9} Q = {0-2-7} P =1-5 Q =? f = 2 1 1 2 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 1 5

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SLIDE 67

Finding spanning path system

P = {0-6, 1-5, 4-8, 7, 9} Q = {0-2-7} P =1-5 Q =4-8 f = 2 1 1 2 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 1 5 4 8

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SLIDE 68

Finding spanning path system

P = {0-6, 1-5, 4-8, 7, 9} Q = {0-2-7} P =1-5 Q =4-8 f = 2 1 1 2 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 1 5 4 8

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SLIDE 69

Finding spanning path system

P = {0-6, 7, 9} Q = {0-2-7, 1-5-8-4} P =? Q =? f = 2 1 1 2 1 1 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 1 5 8 4

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SLIDE 70

Finding spanning path system

P = {0-6, 7, 9} Q = {0-2-7, 1-5-8-4} P =0-6 Q =? f = 2 1 1 2 1 1 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 1 5 8 4 6

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SLIDE 71

Finding spanning path system

P = {0-6, 7, 9} Q = {0-2-7, 1-5-8-4} P =0-6 Q =9 f = 2 1 1 2 1 1 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 1 5 8 4 6 9

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SLIDE 72

Finding spanning path system

P = {0-6, 7, 9} Q = {0-2-7, 1-5-8-4} P =0-6 Q =9 f = 2 1 1 2 1 1 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 1 5 8 4 6 9

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SLIDE 73

Finding spanning path system

P = {0-6-9, 7} Q = {0-2-7, 1-5-8-4} P =? Q =? f = 2 1 1 2 1 1 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 1 5 8 4

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SLIDE 74

Finding spanning path system

P = {0-6-9, 7} Q = {0-2-7, 1-5-8-4} P =7 Q =? f = 2 1 1 2 1 1 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 1 5 8 4 7

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SLIDE 75

Finding spanning path system

P = {0-6-9, 7} Q = {0-2-7, 1-5-8-4} P =7 Q =0-6-9 f = 2 1 1 2 1 1 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 1 5 8 4 7 6 9

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SLIDE 76

Finding spanning path system

P = {0-6-9, 7} Q = {0-2-7, 1-5-8-4} P =7 Q =0-6-9 f = 2 1 1 2 1 1 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 1 5 8 4 7 6 9

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SLIDE 77

Finding spanning path system

P = {} Q = {0-2-7, 1-5-8-4, 0-6-9-7} P =? Q =? f = 2 1 1 2 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 2 7 1 5 8 4 6 9 7

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SLIDE 78

A monotonicity lemma

Assume that the algorithm has been running successfully to produce the path family Pt with |EPt| = t for some nonnegative integer t. Let Σt be the set of finished vertices for Pt. Denote by Ωt the family of paths containing no fresh vertices in the path family Pt. If π1 ∈ VH \ f −1(0), then |Σt| + |Ωt| ≤ |Σt−1| + |Ωt−1| ≤ · · · ≤ |Σ0| + |Ω0| = |f −1(0)| +

v∈VH f(v).

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SLIDE 79

Monotonicity of |Σt| + |Ωt|

t = 0 E = (0-2, 0-6, 1-5, 2-7, 4-8, 5-8, 6-9, 7-9) Σt = {3} Ωt = {0, 0, 1, 4, 7, 7} |Σt| + |Ωt| = 7 f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3

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SLIDE 80

Monotonicity of |Σt| + |Ωt|

t = 1 E = (0-2, 0-6, 1-5, 2-7, 4-8, 5-8, 6-9, 7-9) Σt = {3} Ωt = {0, 0-2, 1, 4, 7, 7} |Σt| + |Ωt| = 7 f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3

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SLIDE 81

Monotonicity of |Σt| + |Ωt|

t = 2 E = (0-2, 0-6, 1-5, 2-7, 4-8, 5-8, 6-9, 7-9) Σt = {3} Ωt = {0-2, 0-6, 1, 4, 7, 7} |Σt| + |Ωt| = 7 f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 2

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SLIDE 82

Monotonicity of |Σt| + |Ωt|

t = 3 E = (0-2, 0-6, 1-5, 2-7, 4-8, 5-8, 6-9, 7-9) Σt = {3} Ωt = {0-2, 0-6, 1-5, 4, 7, 7} |Σt| + |Ωt| = 7 f = 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 2 6

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SLIDE 83

Monotonicity of |Σt| + |Ωt|

t = 4 E = (0-2, 0-6, 1-5, 2-7, 4-8, 5-8, 6-9, 7-9) Σt = {3} Ωt = {0-2-7, 0-6, 1-5, 4, 7} |Σt| + |Ωt| = 6 f = 2 1 1 2 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 6 5

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SLIDE 84

Monotonicity of |Σt| + |Ωt|

t = 5 E = (0-2, 0-6, 1-5, 2-7, 4-8, 5-8, 6-9, 7-9) Σt = {} Ωt = {0-2-7, 0-6, 1-5, 4-8, 7} |Σt| + |Ωt| = 5 f = 2 1 1 2 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 6 5 7 3

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SLIDE 85

Monotonicity of |Σt| + |Ωt|

t = 6 E = (0-2, 0-6, 1-5, 2-7, 4-8, 5-8, 6-9, 7-9) Σt = {8} Ωt = {0-2-7, 0-6, 1-5-8-4, 7} |Σt| + |Ωt| = 5 f = 2 1 1 2 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 6 5 7 3 8

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SLIDE 86

Monotonicity of |Σt| + |Ωt|

t = 7 E = (0-2, 0-6, 1-5, 2-7, 4-8, 5-8, 6-9, 7-9) Σt = {8} Ωt = {0-2-7, 0-6-9, 1-5-8-4, 7} |Σt| + |Ωt| = 5 f = 2 1 1 2 1 1 1 1 1 4 7 3 2 5 6 8 9 1 4 7 3 2 6 5 7 3 8

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SLIDE 87

Monotonicity of |Σt| + |Ωt|

t = 8 E = (0-2, 0-6, 1-5, 2-7, 4-8, 5-8, 6-9, 7-9) Σt = {8, 9} Ωt = {0-2-7, 0-6-9-7, 1-5-8-4} |Σt| + |Ωt| = 5 f = 2 1 1 2 2 1 1 2 1 4 7 3 2 5 6 8 9 1 4 7 3 2 6 5 7 3 8 9 8 9

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SLIDE 88

The end

Thank you

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SLIDE 89

Tolerating edge failures

We can increase the Hamiltonian thickness bound by one to deal with every edge failure. v u v x u x v u

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