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

Polona Pavliˇ c1 Janez ˇ Zerovnik1,2

1IMFM, Ljubljana, Slovenia 2FME, University of Ljubljana, Slovenia

Ljubljana-Leoben 2012

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

Polygraphs

• Definition. A polygraph Ωn = Ωn(G1, . . . Gn; X1, . . . Xn) over

mutually disjoint monographs G1, . . . , Gn has the vertex set V (Ωn) = V (G1) ∪ . . . ∪ V (Gn), and the edge set E(Ωn) = E(G1) ∪ X1 ∪ . . . ∪ E(Gn) ∪ Xn, where Xi ⊆ V (Gi) × V (Gi+1) for i = 1, . . . , n and Gn+1 ∼ = G1.

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

G1 G2 G3 G4 G5

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

G1 G2 G3 G4 G5 X1 X2 X3 X4 X5

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

G1 G2 G3 G4 G5 X1 X2 X3 X4 X5 Di, i = 1, . . . , 5

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

G1 G2 G3 G4 G5 X1 X2 X3 X4 X5 Di, i = 1, . . . , 5 Ri, i = 1, . . . , 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

P12P5

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

P12P5 C12P5

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

P12 ⊠ P5 P12 × P5

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),...

• ther 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.

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

Examples of path algebras: P1 = ({0, 1} , max, min, 0, 1) P2 = (N0 ∪ {−∞}, max, +, −∞, 0) P3 = (N0 ∪ {∞}, min, +, ∞, 0)

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

Examples of path algebras: P1 = ({0, 1} , max, min, 0, 1) P2 = (N0 ∪ {−∞}, max, +, −∞, 0) P3 = (N0 ∪ {∞}, min, +, ∞, 0) ”tropical semiring”

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

Matrices with elements of a path algebra: Let P = (P, ⊕, ◦, e⊕, e◦) be a path algebra and let Mn(P) be the set

• f all n × n matrices over P. Let A, B ∈ Mn(P) and define operations

⊕ and ◦ in the usual way: (A ⊕ B)ij = Aij ⊕ Bij, (A ◦ B)ij =

n

• k=1

Aik ◦ Bkj. Observation: Mn(P) equipped with above operations is a path algebra with the zero and the unit matrix as units of semiring.

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

Matrices with elements of a path algebra: Let P = (P, ⊕, ◦, e⊕, e◦) be a path algebra and let Mn(P) be the set

• f all n × n matrices over P. Let A, B ∈ Mn(P) and define operations

⊕ and ◦ in the usual way: (A ⊕ B)ij = Aij ⊕ Bij, (A ◦ B)ij =

n

• k=1

Aik ◦ Bkj. Observation: Mn(P) equipped with above operations is a path algebra with the zero and the unit matrix as units of semiring.

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

Example: P = (N0 ∪ {∞}, min, +, ∞, 0). The zero matrix:    ∞ ∞ . . . ∞ . . . . . . ... . . . ∞ ∞ . . . ∞    The unit matrix:      ∞ . . . ∞ ∞ . . . ∞ . . . . . . ... . . . ∞ ∞ . . .      (A ⊕ B)ij = min {Aij, Bij}, (A ◦ B)ij = mink=1,...,n {Aik + Bkj}

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

Path algebras and digraphs

Let P = (P, ⊕, ◦, e⊕, e◦) be a path algebra and let G be a labeled digraph, that is a digraph together with a labeling function ℓ which assigns to every arc of G an element of P: ℓ : E(G) − → P. Let V (G) = {v1, v2, . . . , vn} and define the following matrix: (A (G))ij =

• ℓ (vi, vj) ;

if (vi, vj) is an arc of G e⊕;

• therwise

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

The labeling ℓ of G can be extended to walks: for a walk Q = (vi0, vi1)(vi1, vi2) . . . (vik−1, vik) of G let ℓ(Q) = ℓ (vi0, vi1) ◦ ℓ (vi1, vi2) ◦ . . . ◦ ℓ

• vik−1, vik
• .

Observation: Let Sk

ij be the set of all walks of order k from vi to vj

in G. Then

• A(G)k

ij =

• Q∈Sk

ij

ℓ(Q).

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

The labeling ℓ of G can be extended to walks: for a walk Q = (vi0, vi1)(vi1, vi2) . . . (vik−1, vik) of G let ℓ(Q) = ℓ (vi0, vi1) ◦ ℓ (vi1, vi2) ◦ . . . ◦ ℓ

• vik−1, vik
• .

Observation: Let Sk

ij be the set of all walks of order k from vi to vj

in G. Then

• A(G)k

ij =

• Q∈Sk

ij

ℓ(Q).

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

Algorithm, different graph invariants

O(log n) algorithm (Klavˇ zar and ˇ Zerovnik, 1996) Let ωn(G; X) be a rotagraph and ψn(G; X) a fasciagraph. Define a labeled digraph G = G(G; X): V (G) . . . subsets of D ⊔ R; E(G) . . . between vertices that are not in ”conflict”. G1 G2 Vi D1 R1

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

Algorithm, different graph invariants

O(log n) algorithm (Klavˇ zar and ˇ Zerovnik 1996) Let ωn(G; X) be a rotagraph and ψn(G; X) a fasciagraph. Define a labeled digraph G = G(G; X): V (G) . . . subsets of R ⊔ D; E(G) . . . between vertices that are not in ”conflict”.

1 Select appropriate path algebra and define labeling:

ℓ : E(G) − → P.

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

Algorithm, different graph invariants

O(log n) algorithm (Klavˇ zar and ˇ Zerovnik, 1996) Let ωn(G; X) be a rotagraph and ψn(G; X) a fasciagraph. Define a labeled digraph G = G(G; X): V (G) . . . subsets of R ⊔ D; E(G) . . . between vertices that are not in ”conflict”.

1 Select appropriate path algebra and define labeling:

ℓ : E(G) − → P.

2 Form A (G) and in M(P) calculate A (G)n.

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

Algorithm, different graph invariants

O(log n) algorithm (Klavˇ zar and ˇ Zerovnik, 1996) Let ωn(G; X) be a rotagraph and ψn(G; X) a fasciagraph. Define a labeled digraph G = G(G; X): V (G) . . . subsets of R ⊔ D; E(G) . . . between vertices that are not in ”conflict”.

1 Select appropriate path algebra and define labeling:

ℓ : E(G) − → P.

2 Form A (G) and in M(P) calculate A (G)n. 3 Among admissable coefficients of A (G)n select one that

• ptimizes the corresponding goal function.

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

Algorithm, different graph invariants

O(log n) algorithm (Klavˇ zar and ˇ Zerovnik, 1996) Let ωn(G; X) be a rotagraph and ψn(G; X) a fasciagraph. Define a labeled digraph G = G(G; X): V (G) . . . subsets of R ⊔ D; E(G) . . . between vertices that are not in ”conflict”.

1 Select appropriate path algebra and define labeling:

ℓ : E(G) − → P.

2 Form A (G) and in M(P) calculate A (G)n. ←

− O(log n)

3 Among admissable coefficients of A (G)n select one that

• ptimizes the corresponding goal function.

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

The domination number: (Klavˇ zar, ˇ Zerovnik,1996)

1 G = G(G; X):

V (G) = {Vi | Vi ⊆ R ⊔ D};

2 P = (N0 ∪ {∞}, min, +, ∞, 0). 3 ℓ(Vi, Vj) = |Vi ∩ R| + γi,j(G; X) + |D ∩ Vj| − |Vi ∩ R ∩ D ∩ Vj| .

G1 G2 G3 Vi Vj D1 D2 R1 R2

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

The domination number: (Klavˇ zar, ˇ Zerovnik,1996)

1 G = G(G; X):

V (G) = {Vi | Vi ⊆ R ⊔ D};

2 P = (N0 ∪ {∞}, min, +, ∞, 0). 3 ℓ(Vi, Vj) = |Vi ∩ R| + γi,j(G; X) + |D ∩ Vj| − |Vi ∩ R ∩ D ∩ Vj| .

G1 G2 G3 Vi Vj D1 D2 R1 R2

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

The domination number: (Klavˇ zar, ˇ Zerovnik,1996)

1 G = G(G; X):

V (G) = {Vi | Vi ⊆ R ⊔ D};

2 P = (N0 ∪ {∞}, min, +, ∞, 0). 3 ℓ(Vi, Vj) = |Vi ∩ R| + γi,j(G; X) + |D ∩ Vj| − |Vi ∩ R ∩ D ∩ Vj| .

Then γ (ψn(G; X)) = (A (G)n)00 and γ (ωn(G; X)) = min

i

(A (G)n)ii .

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

Generalization

Time complexity;

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

Generalization

Time complexity; Space complexity;

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

Generalization

Time complexity; Space complexity; Other domination - type invariants;

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

Generalization

Time complexity; Space complexity; Other domination - type invariants; Implementation to get some closed expressions.

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

Calculating A (G)n in O(1)

Let An denote A (G)n and A ∈ Mn(P), where P = (N0 ∪ {∞} , min, +, ∞, 0). Lemma (-, ˇ Zerovnik, 2012) Let N = |V (G(G; X))|, K = |V (G)|. Then there is an index q such that Aq = Ap + C for some index p < q and some constant matrix C = [c]ij. Let P = q − p. Then for every r ≥ p and every s ≥ 0 we have Ar+sP = Ar + sC .

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

Fasciagraphs

γ (ψn(G; X)) = (A (G)n)00 An

0i = min k

• An−1

0k

+ Aki

• .

Lemma (-, ˇ Zerovnik, 2012) Assume that the j–th row of An+P and An differ for a constant, an+P

ji

= an

ji + C for all i. Then mini an+P ji

= mini an

ji + C.

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

Fasciagraphs

γ (ψn(G; X)) = (A (G)n)00 An

0i = min k

• An−1

0k

+ Aki

• .

Lemma (-, ˇ Zerovnik, 2012) Assume that the j–th row of An+P and An differ for a constant, an+P

ji

= an

ji + C for all i. Then mini an+P ji

= mini an

ji + C.

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

Fasciagraphs

γ (ψn(G; X)) = (A (G)n)00 An

0i = min k

• An−1

0k

+ Aki

• .

Lemma (-, ˇ Zerovnik, 2012) Assume that the j–th row of An+P and An differ for a constant, an+P

ji

= an

ji + C for all i. Then mini an+P ji

= mini an

ji + C.

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

Rotagraphs

Lemma (-, ˇ Zerovnik, 2012) Let Aq = Ap + C and P = q − p. Then for every t ∈ 0, 1, . . . , P − 1 there is a constant Ct such that for all n ≥ p with t ≡ (n − p) (mod P) we have γ(ψn(G; X)) − γ(ωn(G; X)) = Ct.

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

Results and remarks

Theorem (-, ˇ Zerovnik, 2012) Domination numbers of fasciagraphs and rotagraphs can be computed in constant time, i.e. independently of the size of a monograph G.

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks k γ(PnCk) 3

• 3n

4

• + 1;

if n ≡ 0 (mod 4)

• 3n

4

• ;
• therwise

4 n 5 3; if n = 2 4; if n = 3 n + 2;

• therwise

6

• 4n

3

• ;

if n ≡ 1 (mod 3)

• 4n

3

• + 1;
• therwise

7

• 3n

2

• + 1;

if n ≡ 1 (mod 2)

• 3n

2

• + 2;
• therwise

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks k γ(PnCk) 8 4; if n = 2 6; if n = 3 8; if n = 4

• 9n

5

• + 1;

if n ≡ 5 (mod 10)

• 9n

5

• + 2;
• therwise

9 5; if n = 2 7; if n = 3 10 if n = 4 2n + 2;

• therwise

10 2n + 2; if n ≤ 5 2n + 3; if 6 ≤ n ≤ 9 2n + 4

• therwise

11

• 19n

8

• + 1;

if n ∈ {1, 2, 4, 6} or n ≡ 3 (mod 8)

• 19n

8

• + 2;
• therwise

Polygraphs Domination and its variations Path algebra The algorithm Results and remarks k γ(CnPk) γ(CnCk) 2

• n

2

• + 1;

if n ≡ 2 (mod 4)

• n

2

• ;
• therwise

3

• 3n

4

• 3n

4

• 4

n + 1; if n ∈ {5, 9} n;

• therwise

n 5 4; if n = 3

• 6n

5

• + 1;

if n ≡ 3, 5, 9 (mod 10)

• 6n

5

• ;
• therwise

n; if n ≡ 0 (mod 5) n + 2; if n ≡ 3 (mod 5) n + 1;

• therwise

6 9; if n = 6

• 10n

7

• + 1;

n ≡ 2, 6, 7, 9 13 (mod 14)

• 10n

7

• ;
• therwise
• 4n

3

• + 1;

n ≡ 2, 3, 8, 9 (mod 18) 11, 14, 15, 17 (mod 18)

• 4n

3

• ;
• therwise

7 6; if n = 3 16; if n = 9 36; if n = 21

• 5n

3

• ;
• therwise
• 3n

2

• ;

n ≡ 0, 5, 9 (mod 14)

• 3n

2

• + 2;

n ≡ 2, 8, 12 (mod 14)

• 3n

2

• + 1;
• therwise