Paired domination in subdivided star-free graphs

Paired domination in subdivided star-free graphs

Paired domination in subdivided star-free graphs Paul Dorbec Nouveau MdC dans l’´ equipe Combalgo, th` eme Graphes et Applications

December 10th, 2009

Paired domination in subdivided star-free graphs Introduction

Domination: an old problem Problem (middle of 19th century) How many queens are needed to threaten all the squares of a chessboard?


Domination: an old problem Problem (middle of 19th century) How many queens are needed to threaten all the squares of a chessboard?


Definitions How many fire stations must be placed to protect every village?


Definitions How many fire stations must be placed to protect every village? −→ γ

D satisfies : ∀u ∈ V , |N[v ] ∩ D| ≥ 1

set of fire stations = covering = dominating set


Definitions How many tips/recycling centers may be placed without irritating the inhabitants of any village? −→ ρ

D satisfies : ∀u ∈ V , |N[v ] ∩ D| ≥ 1 P satisfies : ∀u ∈ V , |N[v ] ∩ P| ≤ 1

set of tips = packing = error correcting code


Duality : |packing | ≤ |dominating set|


Duality : |packing | ≤ |dominating set|


Duality : |packing | ≤ |dominating set| 1

1

1 1 1

1

1 1

1 1 I

1

1

In particular, max(|packing |) = ρ(G ) ≤ γ(G ) = min(|dominating set|).

I

Therefore, if |packing | = |covering |, then both are best possible: these are perfect codes / efficient dominating sets.


Vizing’s Conjecture Conjecture (Vizing, 1968) Given any graphs G and H, γ(G 2H) ≥ γ(G ) × γ(H) I

The other inequality is generally false: γ(P3 ) = 1 and γ(P3 2 P3 ) = 3.


Vizing’s Conjecture Conjecture (Vizing, 1968) Given any graphs G and H, γ(G 2H) ≥ γ(G ) × γ(H) I

The other inequality is generally false: γ(P3 ) = 1 and γ(P3 2 P3 ) = 3.

I

Proved with a factor 2 by Clark and Suen in 2000 : γ(G 2H) ≥ 21 γ(G ) × γ(H)

I

Proved for many graph classes: trees, cycles, chordal graphs... and in some special cases : G 2 G . . .

I

Many similar inequalities for variants of domination, for different products, etc...

Paired domination in subdivided star-free graphs Paired-domination

Definitions Domination: How many fire stations must be placed to protect every village? I

D satisfies : ∀u ∈ V , |N[v ] ∩ D| ≥ 1 −→ γ


Definitions Total domination: What if we also want to protect fire stations? I


I

Dt satisfies : ∀u ∈ V , |N(v ) ∩ D| ≥ 1 −→ γt


Definitions Paired-domination: What about pairing fire-stations? I


I

Dt satisfies : ∀u ∈ V , |N(v ) ∩ D| ≥ 1 −→ γt

I

Dpr dominates G and G [D] has a perfect matching −→ γpr

Paired domination in subdivided star-free graphs Paired-domination Results

Critical case Theorem: [Haynes and Slater (1998)] G connected of order n ≥ 3 ⇒ γpr (G ) ≤ n − 1, with equality iff G = C3 , C5 or a subdivided star ∗ . K1,a c


Critical case Theorem: [Haynes and Slater (1998)] G connected of order n ≥ 3 ⇒ γpr (G ) ≤ n − 1, with equality iff G = C3 , C5 or a subdivided star ∗ . K1,a I

connected, δ ≥ 2 I I I

I

c

n ≥ 6 ⇒ 32 n (Haynes, Slater 98) n ≥ 10 ⇒ 23 (n − 1) (Henning 07) ` ´ girth ≥ 6 ⇒ 23 n − (δ−1)(δ−2) 2 (Chen, Chee Shiu, Hong Shan 08)

connected, cubic, claw-free ⇒ γpr (G ) ≤ (Favaron, Henning 04)

n 2


Critical case Theorem: [Haynes and Slater (1998)] G connected of order n ≥ 3 ⇒ γpr (G ) ≤ n − 1, with equality iff G = C3 , C5 or a subdivided star ∗ . K1,a I

connected, δ ≥ 2 I I I

I

c

n ≥ 6 ⇒ 32 n (Haynes, Slater 98) n ≥ 10 ⇒ 23 (n − 1) (Henning 07) ` ´ girth ≥ 6 ⇒ 23 n − (δ−1)(δ−2) 2 (Chen, Chee Shiu, Hong Shan 08)

connected, cubic, claw-free ⇒ γpr (G ) ≤ (Favaron, Henning 04)

n 2


The bad guys of paired-domination. ∗ A graph G with no induced subdivided star K1,a+2 that satisfies 2(an + 1) γpr (G ) = 2a + 1

Kp+1

a

...

...

...

...

...

...

...

...

...

a

n = p(2a + 1) + 2 γpr (G ) = 2ap + 2

a


Our results Let G be a connected graph of order n ≥ 2. c

Star-free [D., Gravier, Henning (2006)] : If G contains no star K1,a+2 , (a ≥ 0), then γpr (G ) ≤

All these bounds are tight.

2(an+1) 2a+1




2(an+1) 2a+1

P5 -free [D.,Gravier (2008)] : If G is not C5 and contains no induced P5 , then γpr (G ) ≤

All these bounds are tight.

n 2

+1




2(an+1) 2a+1

P5 -free [D.,Gravier (2008)] : If G is not C5 and contains no induced P5 , then γpr (G ) ≤

n 2

+1

Subdivided star-free [D., Gravier (2009?)] : ∗ If G contains no subdivided star K1,a+2 , (a ≥ 1), then γpr (G ) ≤ All these bounds are tight.

2(an+1) 2a+1

Paired domination in subdivided star-free graphs Paired-domination Proof

Preliminary to the proof

Observation If the vertices of a graph G may be partitioned into 2 sets V1 and V2 in such a way that (|V1 | = n1 , |V2 | = n2 )  1 +1)  γpr (G [V1 ]) ≤ 2(an (induction) 2a+1 

γpr (G [V2 ]) ≤

then γpr (G ) ≤

2(an+1) 2a+1 .

2(an2 ) 2a+1

(a little stronger)


Proof

( γpr (G ) ≤

2(an+1) 2a+1 )

By minimum counter-example: I

Let a be the smallest integer for which the inequality is false

I

Let G be a minimum counter-example without subd. star of size a + 2.

F

G

u

? ... C1

C2

Ck


Proof

( γpr (G ) ≤

2(an+1) 2a+1 )



I


I

∗ G contains a subdivided star K1,a+1 , let u be its centre.

F

G

u

... C1

C2

Ck


Proof

( γpr (G ) ≤

2(an+1) 2a+1 )



I


I


I

Let F ⊆ N(u) max s.t. N(F ) = {u}.

u

F

... C1

C2

Ck


Proof

( γpr (G ) ≤

2(an+1) 2a+1 )



I


I


I


I

We have |F | ≤ 1

u

F

... C1

C2

Ck


Proof

( γpr (G ) ≤

2(an+1) 2a+1 )



I


I


I


I

We have |F | ≤ 1

I

Let C1 , . . . , Ck be the components of G [V \ ({u} ∪ F )]

F

u

... C1

C2

Ck


Proof

( γpr (G ) ≤

2(an+1) 2a+1 )



I


I


I


I

We have |F | ≤ 1

I


I

2 ≤ k ≤ a + 1.

F

u

... C1

C2

Ck


Proof

( γpr (G ) ≤

2(an+1) 2a+1 )



I


I


I


I

We have |F | ≤ 1

I


I

2 ≤ k ≤ a + 1.

F

u

...

I

If ∃ Ci such that γpr (Ci ) ≤

C1 2ani 2a+1

C2

Ck


Proof

( γpr (G ) ≤

2(an+1) 2a+1 )



I


I


I


I

We have |F | ≤ 1

I


I

2 ≤ k ≤ a + 1.

F

u

...

I


C1 2ani 2a+1

: OK

C2

Ck


Proof

( γpr (G ) ≤

2(an+1) 2a+1 )



I


I


I


I

We have |F | ≤ 1

I


I

2 ≤ k ≤ a + 1.

F

u

...

I I


C1 2ani 2a+1

Ck

: OK

2(a(n−n1 )+1) 2a+1

Otherwise, γpr (G [V \ C1 ]) ≤ 1) → γpr (G [V \ C1 ]) ≤ 2a(n−n 2 2a+1

C2

=

P

γpr (Ci ) + E, with E < 2

Paired domination in subdivided star-free graphs Conclusion Guess what this is!

Thank you for your attention.

Paired domination in subdivided star-free graphs Conclusion Guess what this is!

Thank you for your attention.

Labrit’s cityhall!