Edge-connectivity of permutation hypergraphs Zolt an Szigeti - - PowerPoint PPT Presentation

Edge-connectivity of permutation hypergraphs

Zolt´ an Szigeti

Laboratoire G-SCOP INP Grenoble, France

29 september 2010

joint work with Neil Jami, Ensimag, INP Grenoble, France

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 1 / 1

Outline

1 Permutation graphs 2 Splitting off in graphs 3 Permutation hypergraphs 4 Splitting off in hypergraphs

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 2 / 1

Permutation graphs

Definition

Given a graph G on n vertices and a permutation π of [n], we define the permutation graph Gπ as follows :

1 we take 2 disjoint copies G1 = (V1, E1) and G2 = (V2, E2) of G, 2 for every vertex vi ∈ V1, we add an edge between vi of G1 and vπ(i)

• f G2, this edge set is denoted by E3,

3 Gπ = (V1 ∪ V2, E1 ∪ E2 ∪ E3).

G π = (123)

G π = (123)

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 3 / 1

Permutation graphs

Definition

Given a graph G on n vertices and a permutation π of [n], we define the permutation graph Gπ as follows :

1 we take 2 disjoint copies G1 = (V1, E1) and G2 = (V2, E2) of G, 2 for every vertex vi ∈ V1, we add an edge between vi of G1 and vπ(i)

• f G2, this edge set is denoted by E3,

3 Gπ = (V1 ∪ V2, E1 ∪ E2 ∪ E3).

G π = (123)

G1 G2

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 3 / 1

Permutation graphs

Definition

Given a graph G on n vertices and a permutation π of [n], we define the permutation graph Gπ as follows :

1 we take 2 disjoint copies G1 = (V1, E1) and G2 = (V2, E2) of G, 2 for every vertex vi ∈ V1, we add an edge between vi of G1 and vπ(i)

• f G2, this edge set is denoted by E3,

3 Gπ = (V1 ∪ V2, E1 ∪ E2 ∪ E3).

G π = (123)

G1 G2 v1 v3 v2 vπ(1) vπ(3) vπ(2)

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 3 / 1

Permutation graphs

Definition

Given a graph G on n vertices and a permutation π of [n], we define the permutation graph Gπ as follows :

1 we take 2 disjoint copies G1 = (V1, E1) and G2 = (V2, E2) of G, 2 for every vertex vi ∈ V1, we add an edge between vi of G1 and vπ(i)

• f G2, this edge set is denoted by E3,

3 Gπ = (V1 ∪ V2, E1 ∪ E2 ∪ E3).

G π = (123)

Gπ

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 3 / 1

Connectivity of permutation graphs

Definition

Edge-connectivity of G : λ(G) = min{dG(X) : ∅ = X ⊂ V }, Minimum degree of G : δ(G) = min{dG(v) : v ∈ V }.

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 4 / 1

Connectivity of permutation graphs

Definition

Edge-connectivity of G : λ(G) = min{dG(X) : ∅ = X ⊂ V }, Minimum degree of G : δ(G) = min{dG(v) : v ∈ V }.

Remark

λ(Gπ) ≤ δ(Gπ) = δ(G) + 1.

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 4 / 1

Connectivity of permutation graphs

Definition

Edge-connectivity of G : λ(G) = min{dG(X) : ∅ = X ⊂ V }, Minimum degree of G : δ(G) = min{dG(v) : v ∈ V }.

Remark

λ(Gπ) ≤ δ(Gπ) = δ(G) + 1.

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1 if and only if G = 2Kk for some odd k.

2K3

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 4 / 1

Necessity

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1 if and only if G = 2Kk for some odd k.

2K3

2K3

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 5 / 1

Necessity

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1 if and only if G = 2Kk for some odd k.

2K3

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 5 / 1

Necessity

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1 if and only if G = 2Kk for some odd k.

2K3

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 5 / 1

Necessity

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1 if and only if G = 2Kk for some odd k.

2K3

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 5 / 1

Sufficiency : Idea

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1 if and only if G = 2Kk for some odd k. G Extension : λ(H) = δ(G) + 1, Splitting off : between G1 and G2, maintaining edge-connectivity

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 6 / 1

Sufficiency : Idea

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1 if and only if G = 2Kk for some odd k. G1 G2 Extension : λ(H) = δ(G) + 1, Splitting off : between G1 and G2, maintaining edge-connectivity

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 6 / 1

Sufficiency : Idea

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1 if and only if G = 2Kk for some odd k. G1 G2 Extension : λ(H) = δ(G) + 1, Splitting off : between G1 and G2, maintaining edge-connectivity

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 6 / 1

Sufficiency : Idea

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1 if and only if G = 2Kk for some odd k. G1 G2 s H Extension : λ(H) = δ(G) + 1, Splitting off : between G1 and G2, maintaining edge-connectivity

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 6 / 1

Sufficiency : Idea

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1 if and only if G = 2Kk for some odd k. G1 G2 Extension : λ(H) = δ(G) + 1, Splitting off : between G1 and G2, maintaining edge-connectivity

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 6 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

H s

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

P s

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

s u v

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

s u v

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

H s

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

s

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

s

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

s

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

s

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

s

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

H′ s

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

H′ s

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs

Given : graph H = (V + s, E), partition P = {P1,P2} of V , integer k.

Definition

Splitting off at s : replacing {su, sv} by uv. Complete splitting off at s : a sequence of splitting off isolating s. it is k-admissible if H′ − s is k-edge-connected. it is P-allowed if the new edges are between P1 and P2.

P s

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 7 / 1

Result on splitting off in graphs

Theorem (Bang-Jensen, Gabow, Jord´ an, Szigeti)

Given : graph H = (V + s, E), partition P = {P1, P2} of V , integer k ≥ 2. There exists a k-admissible P-allowed complete splitting off at s if and

• nly if

H is k-edge-connected in V , d(s, P1) = d(s, P2), H contains no C4-obstacle. C4, k = 3 s

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 8 / 1

C4-obstacle

Definition

A partition {A1, A2, A3, A4} of V is called a C4-obstacle of H if k is odd, each Ai is of degree k, no edge exists between Ai and Ai+2, half of the edges incident to s are incident to P1 ∩ (A1 ∪ A3), half of the edges incident to s are incident to P2 ∩ (A2 ∪ A4).

s

A1 A2 A3 A4

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 9 / 1

C4-obstacle

Definition

A partition {A1, A2, A3, A4} of V is called a C4-obstacle of H if k is odd, each Ai is of degree k, no edge exists between Ai and Ai+2, half of the edges incident to s are incident to P1 ∩ (A1 ∪ A3), half of the edges incident to s are incident to P2 ∩ (A2 ∪ A4).

s

A1 A2 A3 A4 A1 A2 A3 A4

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 9 / 1

C4-obstacle

Definition

A partition {A1, A2, A3, A4} of V is called a C4-obstacle of H if k is odd, each Ai is of degree k, no edge exists between Ai and Ai+2, half of the edges incident to s are incident to P1 ∩ (A1 ∪ A3), half of the edges incident to s are incident to P2 ∩ (A2 ∪ A4).

s

A1 A2 A3 A4 A1 A2 A3 A4 < k

2

< k

2

> k

2

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 9 / 1

Sufficiency

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1, if and only if there exists a k-admissible P-allowed complete splitting off at s in H, if and only if H contains no C4-obstacle, if and only if G = 2Kk for some odd k.

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 10 / 1

Sufficiency

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1, if and only if there exists a k-admissible P-allowed complete splitting off at s in H, if and only if H contains no C4-obstacle, if and only if G = 2Kk for some odd k.

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 10 / 1

Sufficiency

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1, if and only if there exists a k-admissible P-allowed complete splitting off at s in H, if and only if H contains no C4-obstacle, if and only if G = 2Kk for some odd k.

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 10 / 1

Sufficiency

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1, if and only if there exists a k-admissible P-allowed complete splitting off at s in H, if and only if H contains no C4-obstacle, if and only if G = 2Kk for some odd k.

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 10 / 1

Sufficiency

Theorem (Goddard, Raines, Slater)

For a simple graph G without isolated vertices, there exists a permutation π such that λ(Gπ) = δ(G) + 1, if and only if there exists a k-admissible P-allowed complete splitting off at s in H, if and only if H contains no C4-obstacle, if and only if G = 2Kk for some odd k.

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 10 / 1

Hypergraphs

Definition

hypergraph : G = (V , E), V = set of vertices, E = set of hyperedges, subsets of V . G is k-edge-connected if each cut contains at least k hyperedges. G

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 11 / 1

Hypergraphs

Definition

hypergraph : G = (V , E), V = set of vertices, E = set of hyperedges, subsets of V . G is k-edge-connected if each cut contains at least k hyperedges. X V − X ≥ k

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 11 / 1

Permutation hypergraphs

Definition

Given a hypergraph G on n vertices and a permutation π of [n], we define the permutation hypergraph Gπ as follows :

1 we take 2 disjoint copies G1 = (V1, E1) and G2 = (V2, E2) of G, 2 for every vertex vi ∈ V1, we add an edge between vi of G1 and vπ(i) of

G2, this edge set is denoted by E3,

3 Gπ = (V1 ∪ V2, E1 ∪ E2 ∪ E3).

G π = (613425)

Gπ

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 12 / 1

Result on splitting off in hypergraphs

Theorem (Bern´ ath, Grappe, Szigeti)

Given : hypergraph H = (V + s, E), where s is incident only to graph edges, partition P = {P1, P2} of V , integer k. There exists a k-admissible P-allowed complete splitting off at s if and only if H is k-edge-connected in V , dH(s) ≥ 2ω(H − s), dH(s, P1) = dH(s, P2), H contains no C4-obstacle.

s

A1 A2 A3 A4 F

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 13 / 1

Connectivity of permutation hypergraphs

Theorem (Jami, Szigeti)

For a hypergraph G and an integer k ≥ 2, there exists a permutation π such that λ(Gπ) = k if and only if

1 dG(X) ≥ k − |X| for all ∅ = X ⊆ V , 2 G is not composed of two connected components both of k vertices,

k being odd.

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 14 / 1

Connectivity of permutation hypergraphs

Theorem (Jami, Szigeti)

For a hypergraph G and an integer k ≥ 2, there exists a permutation π such that λ(Gπ) = k if and only if

1 dG(X) ≥ k − |X| for all ∅ = X ⊆ V , 2 G is not composed of two connected components both of k vertices,

k being odd.

Remark

Implied by Theorem of Bern´ ath, Grappe, Szigeti. Implies Theorem of Goddard, Raines, Slater : if G is a simple graph G without isolated vertices and k = δ(G) + 1, then

k ≥ 2, 1 is satisfied, 2 is implied by G = 2Kk with k odd.

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 14 / 1

Connectivity of permutation hypergraphs

Theorem (Jami, Szigeti)

For a hypergraph G and an integer k ≥ 2, there exists a permutation π such that λ(Gπ) = k if and only if

1 dG(X) ≥ k − |X| for all ∅ = X ⊆ V , 2 G is not composed of two connected components both of k vertices,

k being odd.

Remark

Implied by Theorem of Bern´ ath, Grappe, Szigeti. Implies Theorem of Goddard, Raines, Slater : if G is a simple graph G without isolated vertices and k = δ(G) + 1, then

k ≥ 2, 1 is satisfied, 2 is implied by G = 2Kk with k odd.

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 14 / 1

Thank you for your attention !

• Z. Szigeti (G-SCOP, Grenoble)

Permutation hypergraphs 29 september 2010 15 / 1