Formulas for various domination numbers of products of paths and - PowerPoint PPT Presentation

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks Formulas for various domination numbers of products of paths and cycles Janez ˇ c 1 Zerovnik 1 , 2 Polona Pavliˇ 1 IMFM, Ljubljana, Slovenia 2 FME, University of Ljubljana, Slovenia Ljubljana-Leoben 2012

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks Polygraphs Definition. A polygraph Ω n = Ω n ( G 1 , . . . G n ; X 1 , . . . X n ) over mutually disjoint monographs G 1 , . . . , G n has the vertex set V (Ω n ) = V ( G 1 ) ∪ . . . ∪ V ( G n ) , and the edge set E (Ω n ) = E ( G 1 ) ∪ X 1 ∪ . . . ∪ E ( G n ) ∪ X n , where X i ⊆ V ( G i ) × V ( G i +1 ) for i = 1 , . . . , n and G n +1 ∼ = G 1 .

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks G 2 G 3 G 4 G 1 G 5

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks G 2 X 2 G 3 X 1 X 3 G 4 G 1 X 4 X 5 G 5

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks G 2 X 2 G 3 X 1 X 3 G 4 G 1 X 4 X 5 D i , i = 1 , . . . , 5 G 5

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks G 2 X 2 G 3 X 1 X 3 G 4 G 1 X 4 X 5 D i , i = 1 , . . . , 5 R i , i = 1 , . . . , 5 G 5

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks Rotagraphs and fasciagraphs

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks Rotagraphs and fasciagraphs

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks Rotagraphs and fasciagraphs

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks Rotagraphs and fasciagraphs

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks P 12 � P 5

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks P 12 � P 5 C 12 � P 5

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks P 12 ⊠ P 5 P 12 × P 5

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks The domination number and its variations Definition. A set D ⊆ V of a graph G = ( V , E ) is a dominating set, if N [ D ] = V . The size of the smallest dominating set of a graph is the domination number, γ ( G ) .

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks The domination number and its variations Definition. A set D ⊆ V of a graph G = ( V , E ) is a dominating set, if N [ D ] = V . The size of the smallest dominating set of a graph is the domination number, γ ( G ) . D is an independent set

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks The domination number and its variations Definition. A set D ⊆ V of a graph G = ( V , E ) is a dominating set, if N [ D ] = V . The size of the smallest dominating set of a graph is the domination number, γ ( G ) . D is an independent set − → the independent domination number ;

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks The domination number and its variations Definition. A set D ⊆ V of a graph G = ( V , E ) is a dominating set, if N [ D ] = V . The size of the smallest dominating set of a graph is the domination number, γ ( G ) . D is an independent set − → the independent domination number ; N ( D ) = V

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks The domination number and its variations Definition. A set D ⊆ V of a graph G = ( V , E ) is a dominating set, if N [ D ] = V . The size of the smallest dominating set of a graph is the domination number, γ ( G ) . D is an independent set − → the independent domination number ; N ( D ) = V − → the total domination number ;

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks The domination number and its variations Definition. A set D ⊆ V of a graph G = ( V , E ) is a dominating set, if N [ D ] = V . The size of the smallest dominating set of a graph is the domination number, γ ( G ) . D is an independent set − → the independent domination number ; N ( D ) = V − → the total domination number ; For every u , v ∈ D , N [ u ] ∩ N [ v ] = ∅

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks The domination number and its variations Definition. A set D ⊆ V of a graph G = ( V , E ) is a dominating set, if N [ D ] = V . The size of the smallest dominating set of a graph is the domination number, γ ( G ) . D is an independent set − → the independent domination number ; N ( D ) = V − → the total domination number ; For every u , v ∈ D , N [ u ] ∩ N [ v ] = ∅ − → the perfect domination number ;

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks The domination number and its variations Definition. A set D ⊆ V of a graph G = ( V , E ) is a dominating set, if N [ D ] = V . The size of the smallest dominating set of a graph is the domination number, γ ( G ) . D is an independent set − → the independent domination number ; N ( D ) = V − → the total domination number ; For every u , v ∈ D , N [ u ] ∩ N [ v ] = ∅ − → the perfect domination number ; Generalizations: the k -domination number;

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks The domination number and its variations Definition. A set D ⊆ V of a graph G = ( V , E ) is a dominating set, if N [ D ] = V . The size of the smallest dominating set of a graph is the domination number, γ ( G ) . D is an independent set − → the independent domination number ; N ( D ) = V − → the total domination number ; For every u , v ∈ D , N [ u ] ∩ N [ v ] = ∅ − → the perfect domination number ; Generalizations: the k -domination number; the Roman domination number;

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks The domination number and its variations Definition. A set D ⊆ V of a graph G = ( V , E ) is a dominating set, if N [ D ] = V . The size of the smallest dominating set of a graph is the domination number, γ ( G ) . D is an independent set − → the independent domination number ; N ( D ) = V − → the total domination number ; For every u , v ∈ D , N [ u ] ∩ N [ v ] = ∅ − → the perfect domination number ; Generalizations: the k -domination number; the Roman domination number; the k -Roman domination number;

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks The domination number and its variations Definition. A set D ⊆ V of a graph G = ( V , E ) is a dominating set, if N [ D ] = V . The size of the smallest dominating set of a graph is the domination number, γ ( G ) . D is an independent set − → the independent domination number ; N ( D ) = V − → the total domination number ; For every u , v ∈ D , N [ u ] ∩ N [ v ] = ∅ − → the perfect domination number ; Generalizations: the k -domination number; the Roman domination number; the k -Roman domination number; the rainbow domination number.

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks Complexity results Theorem. (Johnson, 1979) DOMINATING SET is NP-complete. Proof. Reduction from 3-SAT.

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks Complexity results Theorem. (Johnson, 1979) DOMINATING SET is NP-complete. Proof. Reduction from 3-SAT. Still NP-complete for bipartite graphs (Chang et al., 1984), chordal graphs (Booth and Johnson, 1985),...

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks Complexity results Theorem. (Johnson, 1979) DOMINATING SET is NP-complete. Proof. Reduction from 3-SAT. Still NP-complete for bipartite graphs (Chang et al., 1984), chordal graphs (Booth and Johnson, 1985),... other domination types.

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks Path algebra Definition. A semiring P = ( P , ⊕ , ◦ , e ⊕ , e ◦ ) is a set P on which two binary operations, ⊕ and ◦ are defined such that: 1 ( P , ⊕ ) is a commutative monoid with e ⊕ as a unit; 2 ( P , ◦ ) is a monoid with e ◦ as a unit; 3 ◦ is left– and right–distributive over ⊕ ; 4 for every x ∈ P, x ◦ e ⊕ = e ⊕ = e ⊕ ◦ x An idempotent (x ⊕ x = x for all x ∈ P) semiring is called a path algebra.

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks Path algebra Definition. A semiring P = ( P , ⊕ , ◦ , e ⊕ , e ◦ ) is a set P on which two binary operations, ⊕ and ◦ are defined such that: 1 ( P , ⊕ ) is a commutative monoid with e ⊕ as a unit; 2 ( P , ◦ ) is a monoid with e ◦ as a unit; 3 ◦ is left– and right–distributive over ⊕ ; 4 for every x ∈ P, x ◦ e ⊕ = e ⊕ = e ⊕ ◦ x An idempotent (x ⊕ x = x for all x ∈ P) semiring is called a path algebra.