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

Domination in Hypergraphs Two-way correspondence resulting from Theorem 1 For k ≥ 3 if τ ( H ) ≤ c 1 n H + c 2 m H holds for all H ∈ H k − 1 with − c 2 / ( k − 1 ) < c 1 ≤ c 2 , then γ ( H ) ≤ c 1 n H + ( c 2 − c 1 ) m H holds for all H ∈ H k . 9/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs Two-way correspondence resulting from Theorem 1 For k ≥ 3 if τ ( H ) ≤ c 1 n H + c 2 m H holds for all H ∈ H k − 1 with − c 2 / ( k − 1 ) < c 1 ≤ c 2 , then γ ( H ) ≤ c 1 n H + ( c 2 − c 1 ) m H holds for all H ∈ H k . Moreover, if the former bound is sharp then the latter one is sharp, as well. 9/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs Two-way correspondence resulting from Theorem 1 For k ≥ 3 if 10/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs Two-way correspondence resulting from Theorem 1 For k ≥ 3 if γ ( H ) ≤ an H + bm H 10/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs Two-way correspondence resulting from Theorem 1 For k ≥ 3 if γ ( H ) ≤ an H + bm H holds for all H ∈ H k with real numbers b ≥ 0 and a > − b / k , then 10/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs Two-way correspondence resulting from Theorem 1 For k ≥ 3 if γ ( H ) ≤ an H + bm H holds for all H ∈ H k with real numbers b ≥ 0 and a > − b / k , then τ ( H ) ≤ an H + ( a + b ) m H 10/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs Two-way correspondence resulting from Theorem 1 For k ≥ 3 if γ ( H ) ≤ an H + bm H holds for all H ∈ H k with real numbers b ≥ 0 and a > − b / k , then τ ( H ) ≤ an H + ( a + b ) m H holds for all H ∈ H k − 1 . 10/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs Two-way correspondence resulting from Theorem 1 For k ≥ 3 if γ ( H ) ≤ an H + bm H holds for all H ∈ H k with real numbers b ≥ 0 and a > − b / k , then τ ( H ) ≤ an H + ( a + b ) m H holds for all H ∈ H k − 1 . Moreover, if the former bound is sharp then the latter one is sharp, as well. 10/50 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 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 ) ≤ an H + bm H 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 ) ≤ an H + bm H can be extended to hypergraphs with a less strict condition on 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 ) ≤ an H + bm H can be extended to hypergraphs with a less strict condition on 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 . 11/50 Michael A. Henning Transversals and Domination in Hypergraphs

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 12/50 Michael A. Henning Transversals and Domination in Hypergraphs

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 ) an H + bm H 12/50 Michael A. Henning Transversals and Domination in Hypergraphs

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 ) an H + bm H is the same for H ∈ H + k and for H ∈ H k . 12/50 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. 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 � k � τ ( H ) ≤ n + m 2 . � 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 � k � τ ( H ) ≤ n + m 2 . � 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 � k − 3 � γ ( H ) ≤ n + m 2 . � � 3 ( k − 1 ) 2 13/50 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. 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. 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 , 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 14/50 Michael A. Henning Transversals and Domination in Hypergraphs

Special case when k = 3 � • ❅ � ❅ ✚ ❩ � ❅ ✚ ❩ ✚ ❩ � ❅ ✟✟✟✟✟✟✟ ❡ ❡ • ❅ ❅ ✦ ❧ ❧ ✦ ❅ ❅ ✦ ❧ ❧ ✦ ❅ ✦ ❅ ❧ ❧ ✦ ✦ ❧ ❅ ❧ ❅ ❡ • ❡ ❧ ✱ ❅ � ❧ ✱ ❅ � ❧ ✱ ❅ ❧ ✱ � ❅ • � τ ( H ) = 4 = n + m = 8 + 8 . 4 4 15/50 Michael A. Henning Transversals and Domination in Hypergraphs

Classes of hypergraphs ✐ � ❅ x � ❆ ✁ ❅ � ✁ ❆ ❅ � ✁ ❆ ❅ � ✁ ❆ ❅ ✐ ✁ ✐ ❆ ✐ b 1 b 2 b 3 ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✐ ✁ ✐ ❆ ✐ a 1 a 2 a 3 The hypergraph H 7 . 16/50 Michael A. Henning Transversals and Domination in Hypergraphs

Classes of hypergraphs � ❅ ✐ x 1 � ❅ ❩ ✚ � ❅ ✚ ❩ � ❅ ✚ ❩ ✚ ❩ � ❅ ✟✟✟✟✟✟✟✟✟✟ ✐ ✐ ✐ ❅ ❅ x 2 x 4 x 6 ✦ ❅ ❅ ❧ ❧ ✦ ✦ ❧ ❧ ✦ ❅ ❅ ✦ ❧ ❧ ✦ ✦ ❅ ❅ ❧ ❧ ✦ ✦ ❧ ❅ ❧ ❅ ✐ ✐ ✐ x 3 x 5 x 7 ❧ ✱ ❅ � ❧ ✱ ❅ � ❧ ✱ ❅ � ❧ ✱ ✱ ❧ ❅ � ✐ ❅ � x 8 The hypergraph H 8 . 17/50 Michael A. Henning Transversals and Domination in Hypergraphs

Classes of hypergraphs ✐ u ✏✏✏✏✏ ✏✏✏✏✏ ✜ ✜ ✐ ✐ ✐ y 0 ✜ y 1 ✜ y 2 ❏ ❏ ✜ ✜ ❏ ✜ ❏ ✜ ✜ ❏ ✜ ❏ ✜ ❏ ✜ ❏ ✐ ✐ ✐ ❏ ❏ x 0 x 1 x 2 ❵ ❵ ❵ ❵ ❵ ❵ ❏ ❏ ❵ ❵ ❵ ❵ ❵ ❵ A hypergraph in the family H d = 1 18/50 Michael A. Henning Transversals and Domination in Hypergraphs

Classes of hypergraphs ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✏ ✭ ✭ ✭ ❵ ✏ ❵ ✏ ❵ ❵ ✏ ✡ ❤ ✜ ❤ ✜ ❤ ✜ ❤ ❵ ❵ ✏ y 0 y 1 y 2 y 3 ✏ ✧ ✡ ✜ ✜ ✜ ❡ ❡ ❡ ❏ ✧ ❙ ✪ ❤ ❙ ❤ ✡ ✧ ❡ y 13 y 4 ✜ ✜ ✜ ❏ ❡ ❡ ❡ ✆ ❍❍❍❍❍ ❙ ❊ ❙ ✧ ✪ ✡ ❡ ❙ ✧ � ❏ ❙ ✜ ❡ ✜ ❡ ✜ ❡ � ✪ P ✆ ❊ ✡ P � � ❡ ❤ ❤ ❤ ❤ ❤ P ❏ ✪ ❤ ❡ ❡ ❡ P x 0 ❤ x 1 ❤ x 2 ❤ x 3 x 4 � ✆ ❊ ❵❵❵ � ❳ x 13 ❤ ❤ ❤ P ❤ ❤ ❤ ❤ ❤ ❳ ❳ ❤ ❤ ❤ ❤ ❤ ❡ P ✥✥✥✥✥✥ ❳ ❳ ✪ ❡ ❳ ❳ P ✆ ❊ ❳ ✱ ❡ ❡ ❤ ✱ ❤ ✥ y 12 ✱ ❡ ❳❳❳❳ x 5 ✥ ❊ ❊ ✥ ✱ ❡ ✥ ❤ ❳❳❳❳ ❤ ✱ ✥ P ✥ x 12 ✱ y 5 ❡ ❵ ❤ ❤ ❤ ❤ P ❡ ❤ ❤ ❤ ❤ ❤ ❵ ❤ ❊ ✆ ❤ ❤ ❤ P ❵ ❤ ❤ ❤ ❡ ❍ ✪ P ❡ P ❍ ❤ ❊ P � � ✆ ❤ ❤ ❤ ❤ ❤ ❏ ❡ ❡ ❡ x 6 ✪ P x 11 x 10 x 9 x 8 x 7 ❍ ❡ � � ❊ ✧✧✧✧✧ ✡ ❍ ❍ ✪ ✆ � ❡ ✜ ❡ ✜ ❡ ✜ ❙ � ❏ ❙ ❡ ✡ ❊ ✪ ❙ ✆ ❙ ❡ ❡ ❡ ❏ ✜ ✜ ✜ ❡ ❤ ❤ ✡ ❙ ✪ y 11 ❙ y 6 ✏ ❡ ❡ ❡ ❵ ❏ ✜ ✜ ✜ ✏ ❵ ✡ ❵ ✏ ❵ ✏ ❵ ❤ ❤ ❤ ❤ ❵ ✡ ✏ y 10 ✜ ✭ y 9 ✜ ✭ y 8 ✜ ✭ y 7 ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✏ A hypergraph in the family H cyc 19/50 Michael A. Henning Transversals and Domination in Hypergraphs

Classes of hypergraphs ✏✏✏✏✏ ✏✏✏✏✏ ✏✏✏✏✏ ✜ ✜ ✜ ✐ ✐ ✐ ✐ ✜ z 1 ✜ z 2 ✜ z 3 t 3 ❏ ❏ ❏ ✜ ✜ ✜ ❏ ❏ ❏ ✜ ✜ ✜ PPPPP PPPPP ✜ ❏ ✜ ❏ ✜ ❏ ✜ ❏ ✜ ❏ ✜ ❏ ❭ ❭ ✐ ✐ ✐ ✐ ✐ ✐ ❏ ❏ ❏ ❭ ❭ y 2 y 1 t 1 w 1 w 2 w 3 ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❏ ❏ ❏ ❭ ✡ ❭ ✡ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ✡ ✡ ❭ ❭ ✡ ✡ ❭ ❭ ✡ ❭ ✡ ❭ ✐ ✐ ✐ ✡ ✡ x 2 x 1 t 2 ✥ ✥ ✥ ✥ ✥ ✥ ✡ ✥ ✡ ✥ ✥ ✥ ✥ ✥ A hypergraph in the family H ∗ . 20/50 Michael A. Henning Transversals and Domination in Hypergraphs

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. 21/50 Michael A. Henning Transversals and Domination in Hypergraphs

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 21/50 Michael A. Henning Transversals and Domination in Hypergraphs

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 ∈ { H 7 , H 8 } ∪ H d = 1 ∪ H cyc ∪ H ∗ . 21/50 Michael A. Henning Transversals and Domination in Hypergraphs

Classes of hypergraphs ❙ ✓ ❤ u ❙ ✓ PPPP PPPP PPPP ✂ ✏✏✏✏ ✏✏✏✏ ❇ ❙ ✓ ❇ ✂ ❭ ❭ ❭ ✜ ✜ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❇ ✂ y 3 y 2 y 1 y 0 z 0 z 1 z 2 ❭ ❭ ❭ ✜ ✜ ✡ ✡ ✡ ❏ ❏ ✂ ❇ ❭ ❭ ❭ ✜ ✜ ✡ ✡ ✡ ❏ ❏ ✂ ❇ ❭ ❭ ❭ ✜ ✜ ✡ ✡ ✡ ❏ ❏ ✂ ❇ ✡ ❭ ✡ ❭ ✡ ❭ ✜ ❏ ✜ ❏ ✂ ❤ ✡ ❤ ✡ ❤ ✡ ❤ ❇ ❤ ❏ ❤ ❏ ❤ x 3 x 2 x 1 x 0 w 0 w 1 w 2 ✥ ✥ ✥ ❵ ❵ ✥ ✥ ✥ ❵ ❵ ✡ ✥ ✡ ✥ ✡ ✥ ❵ ❏ ❵ ❏ ✥ ✥ ✥ ❵ ❵ ✥ ✥ ✥ ❵ ❵ A hypergraph in the family H 4edge . 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 ∈ H 4edge . 23/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem. Theorem 6 (MAH, C. L¨ owenstein) 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¨ owenstein) For k = 2 and k ≥ 4 , let H be a connected k-uniform hypergraph on n vertices and m . Then, � k � τ ( H ) ≤ n + m 2 , � 3k � 2 24/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem. Theorem 6 (MAH, C. L¨ owenstein) For k = 2 and k ≥ 4 , let H be a connected k-uniform hypergraph on n vertices and m . Then, � k � τ ( H ) ≤ n + m 2 , � 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. (a) The slug E 8 (b) The snail S 7 (c) The tortoise T 8 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 E k 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 E k is a k -uniform hypergraph on k vertices with exactly one edge. For k ≥ 2 even, a tortoise T k 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 ( T k ) = A ∪ B ∪ C and with E ( T k ) = { e 1 , e 2 , e 3 } , where V ( e 1 ) = A ∪ B , V ( e 2 ) = A ∪ C , and V ( e 3 ) = 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 E k is a k -uniform hypergraph on k vertices with exactly one edge. For k ≥ 2 even, a tortoise T k 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 ( T k ) = A ∪ B ∪ C and with E ( T k ) = { e 1 , e 2 , e 3 } , where V ( e 1 ) = A ∪ B , V ( e 2 ) = A ∪ C , and V ( e 3 ) = B ∪ C . For k ≥ 3 odd, a snail S k 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 ( T k ) = A ∪ B ∪ C ∪ D and with E ( T k ) = { e 1 , e 2 , e 3 } , where V ( e 1 ) = A ∪ B , V ( e 2 ) = A ∪ C , and V ( e 3 ) = B ∪ C ∪ D . 26/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem. Theorem 6 (MAH, C. L¨ owenstein) 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¨ owenstein) 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, � k � τ ( H ) ≤ n + m 2 , � 3k � 2 27/50 Michael A. Henning Transversals and Domination in Hypergraphs

Sharpness of Chv´ atal-McDiarmid Theorem. Theorem 6 (MAH, C. L¨ owenstein) 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, � k � τ ( H ) ≤ n + m 2 , � 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 m H 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 m H 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¨ owenstein) 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 m H 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¨ owenstein) 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, � k − 3 � γ ( H ) ≤ n + m 2 , � � 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 m H 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¨ owenstein) 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, � k − 3 � γ ( H ) ≤ n + m 2 , � � 3 ( k − 1 ) 2 with equality if and only if H = exp ( F ) where F ∈ { E k , S K , T k , T ∗ k } is a slug , snail , tortoise or odd tortoise . 29/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs. exp ( T ∗ exp ( E 4 ) exp ( T 4 ) exp ( S 5 ) 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 H 1 H 2 H 3 H 4 H 5 H 6 H 7 H 8 H 9 H 10 H 11 H 12 H 13 H 14 H 15 32/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H Hypergraphs Let H 1 , H 2 , . . . , H 15 be the fifteen hypergraphs shown in the figure. 33/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H Hypergraphs Let H 1 , H 2 , . . . , H 15 be the fifteen hypergraphs shown in the figure. Let H under be a hypergraph every component of which is isomorphic to a hypergraph H i for some i , 1 ≤ i ≤ 15 . 33/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H Hypergraphs Let H 1 , H 2 , . . . , H 15 be the fifteen hypergraphs shown in the figure. Let H under be a hypergraph every component of which is isomorphic to a hypergraph H i for some i , 1 ≤ i ≤ 15 . Each component of H i we call a unit of H under . 33/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H Hypergraphs Let H 1 , H 2 , . . . , H 15 be the fifteen hypergraphs shown in the figure. Let H under be a hypergraph every component of which is isomorphic to a hypergraph H i for some i , 1 ≤ i ≤ 15 . Each component of H i we call a unit of H under . 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 H 1 , H 2 , . . . , H 15 be the fifteen hypergraphs shown in the figure. Let H under be a hypergraph every component of which is isomorphic to a hypergraph H i for some i , 1 ≤ i ≤ 15 . Each component of H i we call a unit of H under . 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 H 1 H 2 H 3 H 4 H 5 H 6 H 7 H 8 H 9 H 10 H 11 H 12 H 13 H 14 H 15 34/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H Hypergraphs Let H be a hypergraph obtained from H under by adding edges of 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 H under by adding edges of 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 H under by adding edges of 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 = H i 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 H under by adding edges of 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 = H i for some i , 1 ≤ i ≤ 15 . We call the hypergraph H under an underlying hypergraph of H . 35/50 Michael A. Henning Transversals and Domination in Hypergraphs

The Family H Hypergraphs Let H be a hypergraph obtained from H under by adding edges of 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 = H i for some i , 1 ≤ i ≤ 15 . We call the hypergraph H under an underlying hypergraph of 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¨ owenstein ( 2011 ) characterized the hypergraphs 36/50 Michael A. Henning Transversals and Domination in Hypergraphs

Domination in Hypergraphs MAH and Christian L¨ owenstein ( 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¨ owenstein ( 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¨ owenstein ( 2011 ) characterized the hypergraphs with all edges of size at least three and with no isolated vertex that have domination number one-third their order. 36/50 Michael A. Henning Transversals and Domination in Hypergraphs