Transversals and Domination in Hypergraphs Michael A. Henning - - PowerPoint PPT Presentation

Transversals and Domination in Hypergraphs

Michael A. Henning

Department of Mathematics University of Johannesburg

24th Cumberland Conference 12 May 2011

1/50 Michael A. Henning Transversals and Domination in Hypergraphs

Hypergraphs

Hypergraphs Hypergraphs are systems of sets which are conceived as natural extensions of graphs:

2/50 Michael A. Henning Transversals and Domination in Hypergraphs

Hypergraphs

Hypergraphs Hypergraphs are systems of sets which are conceived as natural extensions of graphs: A hypergraph H = (V, E) is a finite set V of elements, called vertices, together with a finite multiset E of arbitrary subsets

• f V, called edges.

2/50 Michael A. Henning Transversals and Domination in Hypergraphs

Hypergraphs

Hypergraphs Hypergraphs are systems of sets which are conceived as natural extensions of graphs: A hypergraph H = (V, E) is a finite set V of elements, called vertices, together with a finite multiset E of arbitrary subsets

• f V, called edges.

A hypergraph is k-uniform if every edge has size k.

2/50 Michael A. Henning Transversals and Domination in Hypergraphs

Hypergraphs

Hypergraphs Hypergraphs are systems of sets which are conceived as natural extensions of graphs: A hypergraph H = (V, E) is a finite set V of elements, called vertices, together with a finite multiset E of arbitrary subsets

• f V, called edges.

A hypergraph is k-uniform if every edge has size k. Every (simple) graph is a 2-uniform hypergraph.

2/50 Michael A. Henning Transversals and Domination in Hypergraphs

Transversals

Transversals A transversal in a hypergraph H is a set of vertices that meets every edge (i.e., has a nonempty intersection with every edge of H).

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Transversals

Transversals A transversal in a hypergraph H is a set of vertices that meets every edge (i.e., has a nonempty intersection with every edge of H). The transversal number τ(H) of H is the minimum number

• f vertices meeting every edge.

3/50 Michael A. Henning Transversals and Domination in Hypergraphs

Transversals

Let H = (V, E) be the hypergraph with V = {u, v, w, x, y} and E = {{v, w, x, y}, {u, v, w}, {u, x, y}}.

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Transversals

Let H = (V, E) be the hypergraph with V = {u, v, w, x, y} and E = {{v, w, x, y}, {u, v, w}, {u, x, y}}.

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❭ ❭ ❭ ❇ ❇ ❇ ❇ ❇ ❇ ✂ ✂ ✂ ✂ ✂ ✂ ✜ ✜ ✜

4/50 Michael A. Henning Transversals and Domination in Hypergraphs

Transversals

Let H = (V, E) be the hypergraph with V = {u, v, w, x, y} and E = {{v, w, x, y}, {u, v, w}, {u, x, y}}.

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❭ ❭ ❭ ❇ ❇ ❇ ❇ ❇ ❇ ✂ ✂ ✂ ✂ ✂ ✂ ✜ ✜ ✜

The set T = {u, v} is a transversal in H.

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Transversals

Let H = (V, E) be the hypergraph with V = {u, v, w, x, y} and E = {{v, w, x, y}, {u, v, w}, {u, x, y}}.

❢

u

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w

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❭ ❭ ❭ ❇ ❇ ❇ ❇ ❇ ❇ ✂ ✂ ✂ ✂ ✂ ✂ ✜ ✜ ✜

The set T = {u, v} is a transversal in H.

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❭ ❭ ❭ ❇ ❇ ❇ ❇ ❇ ❇ ✂ ✂ ✂ ✂ ✂ ✂ ✜ ✜ ✜

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Domination in Hypergraphs

Domination in Hypergraphs A dominating set in a hypergraph H = (V, E) is a subset of vertices D ⊆ V such that for every vertex v ∈ V \ D there exists an edge e ∈ E with v ∈ e and e ∩ D = ∅.

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Domination in Hypergraphs

Domination in Hypergraphs A dominating set in a hypergraph H = (V, E) is a subset of vertices D ⊆ V such that for every vertex v ∈ V \ D there exists an edge e ∈ E with v ∈ e and e ∩ D = ∅. Equivalently, every vertex v ∈ V \ D is adjacent with a vertex in D.

5/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

Domination in Hypergraphs A dominating set in a hypergraph H = (V, E) is a subset of vertices D ⊆ V such that for every vertex v ∈ V \ D there exists an edge e ∈ E with v ∈ e and e ∩ D = ∅. Equivalently, every vertex v ∈ V \ D is adjacent with a vertex in D. The domination number γ(H) of H is the minimum cardinality of a dominating set in H.

5/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

Domination in Hypergraphs A dominating set in a hypergraph H = (V, E) is a subset of vertices D ⊆ V such that for every vertex v ∈ V \ D there exists an edge e ∈ E with v ∈ e and e ∩ D = ∅. Equivalently, every vertex v ∈ V \ D is adjacent with a vertex in D. The domination number γ(H) of H is the minimum cardinality of a dominating set in H. Note that γ(H) ≤ τ(H).

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Domination in Hypergraphs

Let H = (V, E) be the hypergraph with V = {u, v, w, x, y} and E = {{v, w, x, y}, {u, v, w}, {u, x, y}}.

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Domination in Hypergraphs

Let H = (V, E) be the hypergraph with V = {u, v, w, x, y} and E = {{v, w, x, y}, {u, v, w}, {u, x, y}}.

❢

u

❢

w

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v

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x

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y

❭ ❭ ❭ ❇ ❇ ❇ ❇ ❇ ❇ ✂ ✂ ✂ ✂ ✂ ✂ ✜ ✜ ✜

6/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

Let H = (V, E) be the hypergraph with V = {u, v, w, x, y} and E = {{v, w, x, y}, {u, v, w}, {u, x, y}}.

❢

u

❢

w

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v

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❭ ❭ ❭ ❇ ❇ ❇ ❇ ❇ ❇ ✂ ✂ ✂ ✂ ✂ ✂ ✜ ✜ ✜

The set T = {u} is a dominating set in H.

6/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

Let H = (V, E) be the hypergraph with V = {u, v, w, x, y} and E = {{v, w, x, y}, {u, v, w}, {u, x, y}}.

❢

u

❢

w

❢

v

❢

x

❢

y

❭ ❭ ❭ ❇ ❇ ❇ ❇ ❇ ❇ ✂ ✂ ✂ ✂ ✂ ✂ ✜ ✜ ✜

The set T = {u} is a dominating set in H.

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❭ ❭ ❭ ❇ ❇ ❇ ❇ ❇ ❇ ✂ ✂ ✂ ✂ ✂ ✂ ✜ ✜ ✜

6/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

Domination in Hypergraphs For a hypergraph H = (V, E), we let nH = |V| and mH = |E|.

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Domination in Hypergraphs

Domination in Hypergraphs For a hypergraph H = (V, E), we let nH = |V| and mH = |E|. For k ≥ 2 an integer, let Hk denotes the class of all k-uniform hypergraphs H with δ(H) ≥ 1.

7/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

Domination in Hypergraphs For a hypergraph H = (V, E), we let nH = |V| and mH = |E|. For k ≥ 2 an integer, let Hk denotes the class of all k-uniform hypergraphs H with δ(H) ≥ 1.

7/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

Domination in Hypergraphs For a hypergraph H = (V, E), we let nH = |V| and mH = |E|. For k ≥ 2 an integer, let Hk denotes the class of all k-uniform hypergraphs H with δ(H) ≥ 1. Csilla Bujt´ as, MAH and Zsolt Tuza (2010) established the following relationship between domination and transversals in hypergraphs.

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Domination in Hypergraphs

Theorem 1 For every integer k ≥ 3 and for any two reals b ≥ 0 and a > −b/k, the following equality holds:

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Domination in Hypergraphs

Theorem 1 For every integer k ≥ 3 and for any two reals b ≥ 0 and a > −b/k, the following equality holds:

sup

H∈Hk

γ(H) anH + bmH = sup

H∈Hk−1

τ(H) anH + (a + b)mH .

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Domination in Hypergraphs

Two-way correspondence resulting from Theorem 1 For k ≥ 3 if

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Domination in Hypergraphs

Two-way correspondence resulting from Theorem 1 For k ≥ 3 if τ(H) ≤ c1nH + c2mH

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Domination in Hypergraphs

Two-way correspondence resulting from Theorem 1 For k ≥ 3 if τ(H) ≤ c1nH + c2mH holds for all H ∈ Hk−1 with −c2/(k − 1) < c1 ≤ c2, then

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Domination in Hypergraphs

Two-way correspondence resulting from Theorem 1 For k ≥ 3 if τ(H) ≤ c1nH + c2mH holds for all H ∈ Hk−1 with −c2/(k − 1) < c1 ≤ c2, then γ(H) ≤ c1nH + (c2 − c1)mH

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Domination in Hypergraphs

Two-way correspondence resulting from Theorem 1 For k ≥ 3 if τ(H) ≤ c1nH + c2mH holds for all H ∈ Hk−1 with −c2/(k − 1) < c1 ≤ c2, then γ(H) ≤ c1nH + (c2 − c1)mH holds for all H ∈ Hk.

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Domination in Hypergraphs

Two-way correspondence resulting from Theorem 1 For k ≥ 3 if τ(H) ≤ c1nH + c2mH holds for all H ∈ Hk−1 with −c2/(k − 1) < c1 ≤ c2, then γ(H) ≤ c1nH + (c2 − c1)mH holds for all H ∈ Hk. Moreover, if the former bound is sharp then the latter one is sharp, as well.

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Domination in Hypergraphs

Two-way correspondence resulting from Theorem 1 For k ≥ 3 if

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Domination in Hypergraphs

Two-way correspondence resulting from Theorem 1 For k ≥ 3 if γ(H) ≤ anH + bmH

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Domination in Hypergraphs

Two-way correspondence resulting from Theorem 1 For k ≥ 3 if γ(H) ≤ anH + bmH holds for all H ∈ Hk with real numbers b ≥ 0 and a > −b/k, then

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Domination in Hypergraphs

Two-way correspondence resulting from Theorem 1 For k ≥ 3 if γ(H) ≤ anH + bmH holds for all H ∈ Hk with real numbers b ≥ 0 and a > −b/k, then τ(H) ≤ anH + (a + b)mH

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Domination in Hypergraphs

Two-way correspondence resulting from Theorem 1 For k ≥ 3 if γ(H) ≤ anH + bmH holds for all H ∈ Hk with real numbers b ≥ 0 and a > −b/k, then τ(H) ≤ anH + (a + b)mH holds for all H ∈ Hk−1.

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Domination in Hypergraphs

Two-way correspondence resulting from Theorem 1 For k ≥ 3 if γ(H) ≤ anH + bmH holds for all H ∈ Hk with real numbers b ≥ 0 and a > −b/k, then τ(H) ≤ anH + (a + b)mH holds for all H ∈ Hk−1. Moreover, if the former bound is sharp then the latter one is sharp, as well.

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Uniform versus Non-Uniform

Uniform versus Non-Uniform Every valid upper bound on the domination number of k-uniform hypergraphs of the form

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Uniform versus Non-Uniform

Uniform versus Non-Uniform Every valid upper bound on the domination number of k-uniform hypergraphs of the form γ(H) ≤ anH + bmH

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Uniform versus Non-Uniform

Uniform versus Non-Uniform Every valid upper bound on the domination number of k-uniform hypergraphs of the form γ(H) ≤ anH + bmH can be extended to hypergraphs with a less strict condition

• n edge sizes.

11/50 Michael A. Henning Transversals and Domination in Hypergraphs

Uniform versus Non-Uniform

Uniform versus Non-Uniform Every valid upper bound on the domination number of k-uniform hypergraphs of the form γ(H) ≤ anH + bmH can be extended to hypergraphs with a less strict condition

• n edge sizes.

For k ≥ 2, let H+

k denote the class of all hypergraphs H with

δ(H) ≥ 1, in which every edge is of size at least k.

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Domination in Hypergraphs

Theorem: For every integer k ≥ 2 and for any two nonnegative reals a and b with a + b > 0, the supremum of

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Domination in Hypergraphs

Theorem: For every integer k ≥ 2 and for any two nonnegative reals a and b with a + b > 0, the supremum of

γ(H) anH + bmH

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Domination in Hypergraphs

Theorem: For every integer k ≥ 2 and for any two nonnegative reals a and b with a + b > 0, the supremum of

γ(H) anH + bmH

is the same for H ∈ H+

k and for H ∈ Hk.

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The Interplay between Domination and Transversals

Chv´ atal, McDiarmid, Small transversals in hypergraphs. Combinatorica 12 (1992), 19–26.

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The Interplay between Domination and Transversals

Chv´ atal, McDiarmid, Small transversals in hypergraphs. Combinatorica 12 (1992), 19–26. Chv´ atal-McDiarmid Theorem For k ≥ 2, if H is a k-uniform hypergraph on n vertices with m edges, then τ(H) ≤ n + k

2

• m

3k

2

• .

13/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Interplay between Domination and Transversals

Chv´ atal, McDiarmid, Small transversals in hypergraphs. Combinatorica 12 (1992), 19–26. Chv´ atal-McDiarmid Theorem For k ≥ 2, if H is a k-uniform hypergraph on n vertices with m edges, then τ(H) ≤ n + k

2

• m

3k

2

• .

Theorem 2 (Bujt´ as, MAH, Tuza) For k ≥ 3, if H is a hypergraph on n vertices with m edges with all edges of size at least k and with δ(H) ≥ 1, then γ(H) ≤ n + k−3

2

• m
• 3(k−1)

2

• .

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Special case when k = 3

Special case when k = 3 When k = 3, the Chv´ atal-McDiarmid bound was independently discovered by Tuza.

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Special case when k = 3

Special case when k = 3 When k = 3, the Chv´ atal-McDiarmid bound was independently discovered by Tuza.

• Z. Tuza, Covering all cliques of a graph. Discrete Math. 86

(1990), 117–126.

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Special case when k = 3

Special case when k = 3 When k = 3, the Chv´ atal-McDiarmid bound was independently discovered by Tuza.

• Z. Tuza, Covering all cliques of a graph. Discrete Math. 86

(1990), 117–126. Theorem 3 If H is a hypergraph on n vertices and m edges where all edges contain at least three vertices,

14/50 Michael A. Henning Transversals and Domination in Hypergraphs

Special case when k = 3

Special case when k = 3 When k = 3, the Chv´ atal-McDiarmid bound was independently discovered by Tuza.

• Z. Tuza, Covering all cliques of a graph. Discrete Math. 86

(1990), 117–126. Theorem 3 If H is a hypergraph on n vertices and m edges where all edges contain at least three vertices, then τ(H) ≤ n + m 4 .

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Special case when k = 3

• ❡

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❅ ❅ ❅ ❩ ❩ ❩ ✚ ✚ ✚

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• ✱

✱ ✱ ✱ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅ ❧ ❧ ❧ ❧ ✟✟✟✟✟✟✟ ✦ ✦ ✦ ✦ ✦ ✦ ✦ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅

τ(H) = 4 = n + m 4 = 8 + 8 4 .

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Classes of hypergraphs

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❅ ❅ ❅ ❅ ❅ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

• The hypergraph H7.

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Classes of hypergraphs

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x8

❅ ❅ ❅ ❅ ❅ ❩ ❩ ❩ ❩ ✚ ✚ ✚ ✚

• ❅

❅ ❅ ❅ ❅

• ✱

✱ ✱ ✱ ✱ ❧ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅ ❅ ❧ ❧ ❧ ❧ ❧ ✟✟✟✟✟✟✟✟✟✟ ✦ ✦ ✦ ✦ ✦ ✦ ✦ ✦ ✦ ❧ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅ ❅

The hypergraph H8.

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Classes of hypergraphs

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✜ ✜ ✜ ✜ ✜ ✜ ❵ ❵ ❵ ❵ ❵ ❵ ✏✏✏✏✏ ❏ ❏ ❏ ❏ ❏ ❏ ✜ ✜ ✜ ✜ ✜ ✜ ❵ ❵ ❵ ❵ ❵ ❵ ✏✏✏✏✏ ❏ ❏ ❏ ❏ ❏ ❏

A hypergraph in the family Hd=1

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Classes of hypergraphs

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x6

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y6

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y9

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x11

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y12

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❤ ❤ ❤ ❤ ✜ ✜ ✜ ✜ ❡ ❡ ❡ ❡ ✭ ✭ ✭ ✭ ❤ ❤ ❤ ❤ ✜ ✜ ✜ ✜ ❡ ❡ ❡ ❡ ✭ ✭ ✭ ✭ ❤ ❤ ❤ ❤ ✜ ✜ ✜ ✜ ❡ ❡ ❡ ❡ ✭ ✭ ✭ ✭ ❵❵❵

• ✧

✧ ✧ ✧ ✧ ❏ ❏ ❏ ❏

• ❵

❵ ❵ ❵ ❵ ❵ ❡ ❡ ❡ ❳❳❳❳ P P P P P P P ✆ ✆ ✆ ✆ ❳❳❳❳ ❡ ❡ ❡ ❡ ❡ ✱ ✱ ✱ ❙ ❙ ❙ ❊ ❊ ❊ ❊ ❊ ✥ ✥ ✥ ✥ ✥ ✥ ❙ ❙ ❙ ✪ ✪ ✪ ✪ ✪ ❤ ❤ ❤ ❤ ✡ ✡ ✡ ✡ ✡ ❍ ❍ ❍ ❍ ❍ ✏ ✏ ✏ ✏ ✏ ✏ ❤ ❤ ❤ ❤ ✜ ✜ ✜ ✜ ❡ ❡ ❡ ❡ ✭ ✭ ✭ ✭ ❤ ❤ ❤ ❤ ✜ ✜ ✜ ✜ ❡ ❡ ❡ ❡ ✭ ✭ ✭ ✭ ❤ ❤ ❤ ❤ ✜ ✜ ✜ ✜ ❡ ❡ ❡ ❡ ✭ ✭ ✭ ✭ ❵ ❵ ❵

• ✧✧✧✧✧

❏ ❏ ❏ ❏

• ❵

❵ ❵ ❵ ❵ ❵ ❡ ❡ ❡ ❳ ❳ ❳ ❳ P P P P P P P ✆ ✆ ✆ ✆ ❳ ❳ ❳ ❳ ❡ ❡ ❡ ❡ ❡ ✱ ✱ ✱ ❙ ❙ ❙ ❊ ❊ ❊ ❊ ❊ ✥✥✥✥✥✥ ❙ ❙ ❙ ✪ ✪ ✪ ✪ ✪ ❤ ❤ ❤ ❤ ✡ ✡ ✡ ✡ ✡ ❍❍❍❍❍ ✏ ✏ ✏ ✏ ✏ ✏

A hypergraph in the family Hcyc

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Classes of hypergraphs

✐

t2

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t1

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t3

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x1

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y1

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x2

✐

y2

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w1

✐

z1

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w2

✐

z2

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w3

✐

z3

❭ ❭ ❭ ❭ ❭ ❭ ✥ ✥ ✥ ✥ ✥ ✥ PPPPP ✡ ✡ ✡ ✡ ✡ ✡ ❭ ❭ ❭ ❭ ❭ ❭ ✥ ✥ ✥ ✥ ✥ ✥ PPPPP ✡ ✡ ✡ ✡ ✡ ✡ ✜ ✜ ✜ ✜ ✜ ✜ ❵ ❵ ❵ ❵ ❵ ❵ ✏✏✏✏✏ ❏ ❏ ❏ ❏ ❏ ❏ ✜ ✜ ✜ ✜ ✜ ✜ ❵ ❵ ❵ ❵ ❵ ❵ ✏✏✏✏✏ ❏ ❏ ❏ ❏ ❏ ❏ ✜ ✜ ✜ ✜ ✜ ✜ ❵ ❵ ❵ ❵ ❵ ❵ ✏✏✏✏✏ ❏ ❏ ❏ ❏ ❏ ❏

A hypergraph in the family H∗.

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Sharpness of Theorem 3.

Sharpness of Theorem 3. MAH, A. Yeo, Hypergraphs with large transversal number and with edge sizes at least three. J. Graph Theory 59 (2008), 326–348.

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Sharpness of Theorem 3.

Sharpness of Theorem 3. MAH, A. Yeo, Hypergraphs with large transversal number and with edge sizes at least three. J. Graph Theory 59 (2008), 326–348. Theorem 4 Let H be a connected 3-uniform hypergraph on n vertices and m edges satisfying τ(H) = n + m 4 .

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Sharpness of Theorem 3.

Sharpness of Theorem 3. MAH, A. Yeo, Hypergraphs with large transversal number and with edge sizes at least three. J. Graph Theory 59 (2008), 326–348. Theorem 4 Let H be a connected 3-uniform hypergraph on n vertices and m edges satisfying τ(H) = n + m 4 . Then, H ∈ {H7, H8} ∪ Hd=1 ∪ Hcyc ∪ H∗.

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Classes of hypergraphs

❤

u

❤

x0

❤

y0

❤

x1

❤

y1

❤

x2

❤

y2

❤

x3

❤

y3

❤

w0

❤

z0

❤

w1

❤

z1

❤

w2

❤

z2

❙ ❙ ❙ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✓ ✓ ✓ ❭ ❭ ❭ ❭ ❭ ✥ ✥ ✥ ✥ ✥ PPPP ✡ ✡ ✡ ✡ ✡ ✡ ❭ ❭ ❭ ❭ ❭ ✥ ✥ ✥ ✥ ✥ PPPP ✡ ✡ ✡ ✡ ✡ ✡ ❭ ❭ ❭ ❭ ❭ ✥ ✥ ✥ ✥ ✥ PPPP ✡ ✡ ✡ ✡ ✡ ✡ ✜ ✜ ✜ ✜ ✜ ❵ ❵ ❵ ❵ ❵ ✏✏✏✏ ❏ ❏ ❏ ❏ ❏ ❏ ✜ ✜ ✜ ✜ ✜ ❵ ❵ ❵ ❵ ❵ ✏✏✏✏ ❏ ❏ ❏ ❏ ❏ ❏

A hypergraph in the family H4edge.

22/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Theorem 3.

Sharpness of Theorem 3. MAH, A. Yeo, J. Graph Theory 59 (2008), 326–348.

23/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Theorem 3.

Sharpness of Theorem 3. MAH, A. Yeo, J. Graph Theory 59 (2008), 326–348. Theorem 5 Let H be a connected hypergraph on n vertices and m edges where all edges contain at least three vertices.

23/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Theorem 3.

Sharpness of Theorem 3. MAH, A. Yeo, J. Graph Theory 59 (2008), 326–348. Theorem 5 Let H be a connected hypergraph on n vertices and m edges where all edges contain at least three vertices. If H is not 3-uniform and τ(H) = n + m 4 ,

23/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Theorem 3.

Sharpness of Theorem 3. MAH, A. Yeo, J. Graph Theory 59 (2008), 326–348. Theorem 5 Let H be a connected hypergraph on n vertices and m edges where all edges contain at least three vertices. If H is not 3-uniform and τ(H) = n + m 4 , then H ∈ H4edge.

23/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem.

Theorem 6 (MAH, C. L¨

• wenstein)

For k = 2 and k ≥ 4, let H be a connected k-uniform hypergraph on n vertices and m.

24/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem.

Theorem 6 (MAH, C. L¨

• wenstein)

For k = 2 and k ≥ 4, let H be a connected k-uniform hypergraph on n vertices and m. Then, τ(H) ≤ n + k

2

• m

3k

2

• ,

24/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem.

Theorem 6 (MAH, C. L¨

• wenstein)

For k = 2 and k ≥ 4, let H be a connected k-uniform hypergraph on n vertices and m. Then, τ(H) ≤ n + k

2

• m

3k

2

• ,

with equality if and only if H is a slug or a snail or a tortoise.

24/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem.

(b) The snail S7 (c) The tortoise T8 (a) The slug E8

25/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem.

Slugs, Snails, and Tortoises For k = 2, a slug Ek is a k-uniform hypergraph on k vertices with exactly one edge.

26/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem.

Slugs, Snails, and Tortoises For k = 2, a slug Ek is a k-uniform hypergraph on k vertices with exactly one edge. For k ≥ 2 even, a tortoise Tk is the k-uniform hypergraph defined as follows. Let A, B and C be vertex-disjoint sets of vertices with |A| = |B| = |C| = k/2. Let V(Tk) = A ∪ B ∪ C and with E(Tk) = {e1, e2, e3}, where V(e1) = A ∪ B, V(e2) = A ∪ C, and V(e3) = B ∪ C.

26/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem.

Slugs, Snails, and Tortoises For k = 2, a slug Ek is a k-uniform hypergraph on k vertices with exactly one edge. For k ≥ 2 even, a tortoise Tk is the k-uniform hypergraph defined as follows. Let A, B and C be vertex-disjoint sets of vertices with |A| = |B| = |C| = k/2. Let V(Tk) = A ∪ B ∪ C and with E(Tk) = {e1, e2, e3}, where V(e1) = A ∪ B, V(e2) = A ∪ C, and V(e3) = B ∪ C. For k ≥ 3 odd, a snail Sk is the k-uniform hypergraph defined as follows. Let A, B, C and D be vertex-disjoint sets of vertices with |A| = (k + 1)/2, |B| = |C| = k/2 and |D| = 1. Let V(Tk) = A ∪ B ∪ C ∪ D and with E(Tk) = {e1, e2, e3}, where V(e1) = A ∪ B, V(e2) = A ∪ C, and V(e3) = B ∪ C ∪ D.

26/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem.

Theorem 6 (MAH, C. L¨

• wenstein)

For k = 2 and k ≥ 4, let H be a connected hypergraph on n vertices and m with all edges of size at least k.

27/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem.

Theorem 6 (MAH, C. L¨

• wenstein)

For k = 2 and k ≥ 4, let H be a connected hypergraph on n vertices and m with all edges of size at least k. Then, τ(H) ≤ n + k

2

• m

3k

2

• ,

27/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem.

Theorem 6 (MAH, C. L¨

• wenstein)

For k = 2 and k ≥ 4, let H be a connected hypergraph on n vertices and m with all edges of size at least k. Then, τ(H) ≤ n + k

2

• m

3k

2

• ,

with equality if and only if H is a slug or a snail or a tortoise or an odd tortoise.

27/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem.

The odd tortoise T ∗

7

28/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs.

Expansion of a Hypergraph The expansion, exp(H), of a hypergraph H is obtained from H by adding mH new vertices, one vertex to each edge of H, so that the added vertices have degree 1 in exp(H).

29/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs.

Expansion of a Hypergraph The expansion, exp(H), of a hypergraph H is obtained from H by adding mH new vertices, one vertex to each edge of H, so that the added vertices have degree 1 in exp(H). Theorem 7 (MAH, C. L¨

• wenstein)

For k ≥ 5, let H be a connected hypergraph on n vertices and m with all edges of size at least k and with δ(H) ≥ 1.

29/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs.

Expansion of a Hypergraph The expansion, exp(H), of a hypergraph H is obtained from H by adding mH new vertices, one vertex to each edge of H, so that the added vertices have degree 1 in exp(H). Theorem 7 (MAH, C. L¨

• wenstein)

For k ≥ 5, let H be a connected hypergraph on n vertices and m with all edges of size at least k and with δ(H) ≥ 1. Then, γ(H) ≤ n + k−3

2

• m
• 3(k−1)

2

• ,

29/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs.

Expansion of a Hypergraph The expansion, exp(H), of a hypergraph H is obtained from H by adding mH new vertices, one vertex to each edge of H, so that the added vertices have degree 1 in exp(H). Theorem 7 (MAH, C. L¨

• wenstein)

For k ≥ 5, let H be a connected hypergraph on n vertices and m with all edges of size at least k and with δ(H) ≥ 1. Then, γ(H) ≤ n + k−3

2

• m
• 3(k−1)

2

• ,

with equality if and only if H = exp(F) where F ∈ {Ek, SK, Tk, T∗

k} is a slug, snail, tortoise or odd tortoise.

29/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs.

exp(E4) exp(T4) exp(S5) exp(T ∗

5 )

30/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

Theorem 8 If H is a hypergraph of order n with all edges of size at least three and with no isolated vertex, then

31/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

Theorem 8 If H is a hypergraph of order n with all edges of size at least three and with no isolated vertex, then γ(H) ≤ n/3.

31/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

Theorem 8 If H is a hypergraph of order n with all edges of size at least three and with no isolated vertex, then γ(H) ≤ n/3. Question What are the hypergraphs achieving equality in the bound of Theorem 8?

31/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15

32/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H

Hypergraphs Let H1, H2, . . . , H15 be the fifteen hypergraphs shown in the figure.

33/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H

Hypergraphs Let H1, H2, . . . , H15 be the fifteen hypergraphs shown in the figure. Let Hunder be a hypergraph every component of which is isomorphic to a hypergraph Hi for some i, 1 ≤ i ≤ 15.

33/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H

Hypergraphs Let H1, H2, . . . , H15 be the fifteen hypergraphs shown in the figure. Let Hunder be a hypergraph every component of which is isomorphic to a hypergraph Hi for some i, 1 ≤ i ≤ 15. Each component of Hi we call a unit of Hunder.

33/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H

Hypergraphs Let H1, H2, . . . , H15 be the fifteen hypergraphs shown in the figure. Let Hunder be a hypergraph every component of which is isomorphic to a hypergraph Hi for some i, 1 ≤ i ≤ 15. Each component of Hi we call a unit of Hunder. In each unit we 2-color the vertices with the colors black and white as indicated in the figure

33/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H

Hypergraphs Let H1, H2, . . . , H15 be the fifteen hypergraphs shown in the figure. Let Hunder be a hypergraph every component of which is isomorphic to a hypergraph Hi for some i, 1 ≤ i ≤ 15. Each component of Hi we call a unit of Hunder. In each unit we 2-color the vertices with the colors black and white as indicated in the figure and we call the white vertices the link vertices of the unit and the black vertices the non-link vertices

33/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15

34/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H

Hypergraphs Let H be a hypergraph obtained from Hunder by adding edges

• f size at least three, called link edges,

35/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H

Hypergraphs Let H be a hypergraph obtained from Hunder by adding edges

• f size at least three, called link edges, in such a way that

every added edge contains vertices from at least two units and contains only link vertices.

35/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H

Hypergraphs Let H be a hypergraph obtained from Hunder by adding edges

• f size at least three, called link edges, in such a way that

every added edge contains vertices from at least two units and contains only link vertices. Possibly, H is disconnected or H = Hi for some i, 1 ≤ i ≤ 15.

35/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H

Hypergraphs Let H be a hypergraph obtained from Hunder by adding edges

• f size at least three, called link edges, in such a way that

every added edge contains vertices from at least two units and contains only link vertices. Possibly, H is disconnected or H = Hi for some i, 1 ≤ i ≤ 15. We call the hypergraph Hunder an underlying hypergraph

• f H.

35/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H

Hypergraphs Let H be a hypergraph obtained from Hunder by adding edges

• f size at least three, called link edges, in such a way that

every added edge contains vertices from at least two units and contains only link vertices. Possibly, H is disconnected or H = Hi for some i, 1 ≤ i ≤ 15. We call the hypergraph Hunder an underlying hypergraph

• f H.

Let H denote the family of all such hypergraphs H

35/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

MAH and Christian L¨

• wenstein (2011) characterized the

hypergraphs

36/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

MAH and Christian L¨

• wenstein (2011) characterized the

hypergraphs with all edges of size at least three

36/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

MAH and Christian L¨

• wenstein (2011) characterized the

hypergraphs with all edges of size at least three and with no isolated vertex

36/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

MAH and Christian L¨

• wenstein (2011) characterized the

hypergraphs with all edges of size at least three and with no isolated vertex that have domination number one-third their

• rder.

36/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

MAH and Christian L¨

• wenstein (2011) characterized the

hypergraphs with all edges of size at least three and with no isolated vertex that have domination number one-third their

• rder.

Theorem 9 If H is a hypergraph of order n with all edges of size at least three and with no isolated vertex. Then,

36/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

MAH and Christian L¨

• wenstein (2011) characterized the

hypergraphs with all edges of size at least three and with no isolated vertex that have domination number one-third their

• rder.

Theorem 9 If H is a hypergraph of order n with all edges of size at least three and with no isolated vertex. Then, γ(H) ≤ n/3

36/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs

MAH and Christian L¨

• wenstein (2011) characterized the

hypergraphs with all edges of size at least three and with no isolated vertex that have domination number one-third their

• rder.

Theorem 9 If H is a hypergraph of order n with all edges of size at least three and with no isolated vertex. Then, γ(H) ≤ n/3 with equality if and only if H ∈ H.

36/50 Michael A. Henning Transversals and Domination in Hypergraphs

Total Domination in Graphs

Total Domination in Graphs Let G = (V, E) be a graph without isolated vertices.

37/50 Michael A. Henning Transversals and Domination in Hypergraphs

Total Domination in Graphs

Total Domination in Graphs Let G = (V, E) be a graph without isolated vertices. A set S ⊆ V is a total dominating set of G if every vertex of G is adjacent to a vertex in S.

37/50 Michael A. Henning Transversals and Domination in Hypergraphs

Total Domination in Graphs

Total Domination in Graphs Let G = (V, E) be a graph without isolated vertices. A set S ⊆ V is a total dominating set of G if every vertex of G is adjacent to a vertex in S. The total domination number γt(G) of G is the minimum cardinality of a total dominating set.

37/50 Michael A. Henning Transversals and Domination in Hypergraphs

Total Domination in Graphs

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ❅ ❅ ❅ ✟✟ ✟

• ❍❍

❍ ✁ ✁ ✁ ❆ ❆ ❆

• ❅

❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅

• ❍

❍ ❍ ❅ ❅ ❅ ✟ ✟ ✟ ❆ ❆ ❆ ✁ ✁ ✁

A graph G with γt(G) = 8.

38/50 Michael A. Henning Transversals and Domination in Hypergraphs

Open Neighborhood Hypergraph

Open Neighborhood Hypergraph For a graph G = (V, E), we denote by ONH(G) the open neighborhood hypergraph of G.

39/50 Michael A. Henning Transversals and Domination in Hypergraphs

Open Neighborhood Hypergraph

Open Neighborhood Hypergraph For a graph G = (V, E), we denote by ONH(G) the open neighborhood hypergraph of G. Hence, ONH(G)= (V, C) is the hypergraph with vertex set V and with edge set

39/50 Michael A. Henning Transversals and Domination in Hypergraphs

Open Neighborhood Hypergraph

Open Neighborhood Hypergraph For a graph G = (V, E), we denote by ONH(G) the open neighborhood hypergraph of G. Hence, ONH(G)= (V, C) is the hypergraph with vertex set V and with edge set C = {NG(x) | x ∈ V(G)},

39/50 Michael A. Henning Transversals and Domination in Hypergraphs

Open Neighborhood Hypergraph

Open Neighborhood Hypergraph For a graph G = (V, E), we denote by ONH(G) the open neighborhood hypergraph of G. Hence, ONH(G)= (V, C) is the hypergraph with vertex set V and with edge set C = {NG(x) | x ∈ V(G)}, consisting of the open neighborhoods of vertices in G.

39/50 Michael A. Henning Transversals and Domination in Hypergraphs

Open Neighborhood Hypergraph

Open Neighborhood Hypergraph For a graph G = (V, E), we denote by ONH(G) the open neighborhood hypergraph of G. Hence, ONH(G)= (V, C) is the hypergraph with vertex set V and with edge set C = {NG(x) | x ∈ V(G)}, consisting of the open neighborhoods of vertices in G. Key Observation. For every graph G, γt(G) = τ(ONH(G)) .

39/50 Michael A. Henning Transversals and Domination in Hypergraphs

Open Neighborhood Hypergraph

Open Neighborhood Hypergraph For a graph G = (V, E), we denote by ONH(G) the open neighborhood hypergraph of G. Hence, ONH(G)= (V, C) is the hypergraph with vertex set V and with edge set C = {NG(x) | x ∈ V(G)}, consisting of the open neighborhoods of vertices in G. Key Observation. For every graph G, γt(G) = τ(ONH(G)) . Thus total domination in graphs can be translated to the problem of finding transversals in hypergraphs.

39/50 Michael A. Henning Transversals and Domination in Hypergraphs

Open Neighborhood Hypergraph

✉

x6

✉

y2

✉

y3

✉

x1

✉

y1

✉

x7

✉

x3

✉

y6

✉

y7

✉

x5

✉

x8

✉

y4

✉

x2

✉

y8

✉

y5

✉

x4

❅ ❅ ❅ ✟✟ ✟

• ❍❍

❍ ✁ ✁ ✁ ❆ ❆ ❆

• ❅

❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅

• ❍

❍ ❍ ❅ ❅ ❅ ✟ ✟ ✟ ❆ ❆ ❆ ✁ ✁ ✁

G

40/50 Michael A. Henning Transversals and Domination in Hypergraphs

Open Neighborhood Hypergraph

✉

x6

✉

y2

✉

y3

✉

x1

✉

y1

✉

x7

✉

x3

✉

y6

✉

y7

✉

x5

✉

x8

✉

y4

✉

x2

✉

y8

✉

y5

✉

x4

❅ ❅ ❅ ✟✟ ✟

• ❍❍

❍ ✁ ✁ ✁ ❆ ❆ ❆

• ❅

❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅

• ❍

❍ ❍ ❅ ❅ ❅ ✟ ✟ ✟ ❆ ❆ ❆ ✁ ✁ ✁

G

41/50 Michael A. Henning Transversals and Domination in Hypergraphs

Open Neighborhood Hypergraph

❢

x1

❢

x2

❢

x3

❢

x4

❢

x5

❢

x6

❢

x7

❢

x8

❅ ❅ ❅ ❅ ❩ ❩ ❩ ✚ ✚ ✚

• ❅

❅ ❅ ❅

• ✱

✱ ✱ ✱ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅ ❧ ❧ ❧ ❧ ✟✟✟✟✟✟✟ ✦ ✦ ✦ ✦ ✦ ✦ ✦ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅ ❢

y1

❢

y2

❢

y3

❢

y4

❢

y5

❢

y6

❢

y7

❢

y8

❅ ❅ ❅ ❅ ❩ ❩ ❩ ✚ ✚ ✚

• ❅

❅ ❅ ❅

• ✱

✱ ✱ ✱ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅ ❧ ❧ ❧ ❧ ✟✟✟✟✟✟✟ ✦ ✦ ✦ ✦ ✦ ✦ ✦ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅

ONH(G).

42/50 Michael A. Henning Transversals and Domination in Hypergraphs

Open Neighborhood Hypergraph

• ❡

❡ ❡

• ❡
• ❅

❅ ❅ ❅ ❩ ❩ ❩ ✚ ✚ ✚

• ❅

❅ ❅ ❅

• ✱

✱ ✱ ✱ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅ ❧ ❧ ❧ ❧ ✟✟✟✟✟✟✟ ✦ ✦ ✦ ✦ ✦ ✦ ✦ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅ ❡

• ❡

❡

• ❡

❅ ❅ ❅ ❅ ❩ ❩ ❩ ✚ ✚ ✚

• ❅

❅ ❅ ❅

• ✱

✱ ✱ ✱ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅ ❧ ❧ ❧ ❧ ✟✟✟✟✟✟✟ ✦ ✦ ✦ ✦ ✦ ✦ ✦ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅

τ(ONH(G)) = 8.

43/50 Michael A. Henning Transversals and Domination in Hypergraphs

Open Neighborhood Hypergraph

❢

x1

❡ ❡ ❡ ❡ ❢

x5

❢

x6

❢

x8

❅ ❅ ❅ ❅ ❩ ❩ ❩ ✚ ✚ ✚

• ❅

❅ ❅ ❅

• ✱

✱ ✱ ✱ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅ ❧ ❧ ❧ ❧ ✟✟✟✟✟✟✟ ✦ ✦ ✦ ✦ ✦ ✦ ✦ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅ ❡ ❡ ❡ ❡ ❢

y2

❢

y3

❢

y4

❢

y7

❅ ❅ ❅ ❅ ❩ ❩ ❩ ✚ ✚ ✚

• ❅

❅ ❅ ❅

• ✱

✱ ✱ ✱ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅ ❧ ❧ ❧ ❧ ✟✟✟✟✟✟✟ ✦ ✦ ✦ ✦ ✦ ✦ ✦ ❧ ❧ ❧ ❧ ❅ ❅ ❅ ❅

τ(ONH(G)) = 8.

44/50 Michael A. Henning Transversals and Domination in Hypergraphs

Open Neighborhood Hypergraph

✉

x6

✉

y2

✉

y3

✉

x1

✉ ✉ ✉ ✉ ✉

y7

✉

x5

✉

x8

✉

y4

✉ ✉ ✉ ✉ ❅ ❅ ❅ ✟✟ ✟

• ❍❍

❍ ✁ ✁ ✁ ❆ ❆ ❆

• ❅

❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅

• ❍

❍ ❍ ❅ ❅ ❅ ✟ ✟ ✟ ❆ ❆ ❆ ✁ ✁ ✁

G γt(G) = 8.

45/50 Michael A. Henning Transversals and Domination in Hypergraphs

Total Domination in Graphs

Minimum degree at least three As a direct consequence of the Chv´ atal-McDiarmid Theorem when k = 3, or

46/50 Michael A. Henning Transversals and Domination in Hypergraphs

Total Domination in Graphs

Minimum degree at least three As a direct consequence of the Chv´ atal-McDiarmid Theorem when k = 3, or Archdeacon, Ellis-Monaghan, Fischer, Froncek, Lam, Seager, Wei, and Yuster (Some remarks on domination.

• J. Graph Theory 46 (2004), 207–210.)

46/50 Michael A. Henning Transversals and Domination in Hypergraphs

Total Domination in Graphs

Minimum degree at least three As a direct consequence of the Chv´ atal-McDiarmid Theorem when k = 3, or Archdeacon, Ellis-Monaghan, Fischer, Froncek, Lam, Seager, Wei, and Yuster (Some remarks on domination.

• J. Graph Theory 46 (2004), 207–210.)

Theorem 6. If G is a graph of order n with δ(G) ≥ 3, then γt(G) ≤ n 2. The extremal graphs can be deducted from the extremal hypergraphs in the case when k = 3 in the Chv´ atal-McDiarmid Theorem.

46/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Theorem 6.

Sharpness of Theorem 6. MAH, A. Yeo, J. Graph Theory 59 (2008), 326–348.

47/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Theorem 6.

Sharpness of Theorem 6. MAH, A. Yeo, J. Graph Theory 59 (2008), 326–348. Theorem 7. If G is a graph of order n with δ(G) ≥ 3, then γt(G) = n 2 if and only if

47/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Theorem 6.

Sharpness of Theorem 6. MAH, A. Yeo, J. Graph Theory 59 (2008), 326–348. Theorem 7. If G is a graph of order n with δ(G) ≥ 3, then γt(G) = n 2 if and only if

G is the generalized Petersen graph G16 of order 16 or

47/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Theorem 6.

Sharpness of Theorem 6. MAH, A. Yeo, J. Graph Theory 59 (2008), 326–348. Theorem 7. If G is a graph of order n with δ(G) ≥ 3, then γt(G) = n 2 if and only if

G is the generalized Petersen graph G16 of order 16 or G ∈ F1 ∪ F2.

47/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Theorem 6.

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ❅ ❅ ❅ ✟✟ ✟

• ❍❍

❍ ✁ ✁ ✁ ❆ ❆ ❆

• ❅

❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅

• ❍

❍ ❍ ❅ ❅ ❅ ✟ ✟ ✟ ❆ ❆ ❆ ✁ ✁ ✁

The generalized Petersen graph G16 with γt(G16) = 8.

48/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Theorem 6.

s s s s s s s s s s s s s s s s ♣♣♣ ♣♣♣ ✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳ ✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳ ✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳ ✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳

A family F1 of cubic graphs G of order n with γt(G) = n 2.

49/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Theorem 6.

s s s s s s s s s s s s s s s s ♣♣♣ ♣♣♣ ✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳ ✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳ ✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳ ✘✘✘✘✘ ✘ ❳❳❳❳❳ ❳ ✟ ☛ ✟ ☛ ✠ ✡ ✠ ✡

A family F2 of cubic graphs G of order n with γt(G) = n 2.

50/50 Michael A. Henning Transversals and Domination in Hypergraphs