Perfect codes in Generalized Sierpi nski Graphs Aline Parreau - - PowerPoint PPT Presentation

Perfect codes in Generalized Sierpi´ nski Graphs

Aline Parreau

Institut Fourier - Universit´ e de Grenoble - France

Join work with Sylvain Gravier and Matjaz Kovˇ se CID 2011- September 20th, 2011 ANR IDEA

1/21

Outline

Sierpi´ nski Graphs: → Graph on {1, ..., k}n with good metric and coding properties. Idea : generalize those graphs to have new (and good?) metrics on {1, ..., k}n

2/21

Recursive construction of Sierpi´ nski graph S(n, k)

• 1. Start with the complete graph: S(1, k) = Kk.

1 2 3

3/21

Recursive construction of Sierpi´ nski graph S(n, k)

• 1. Start with the complete graph: S(1, k) = Kk.
• 2. Copy it k times.

3/21

Recursive construction of Sierpi´ nski graph S(n, k)

• 1. Start with the complete graph: S(1, k) = Kk.
• 2. Copy it k times.
• 3. Add one edge between each pair of copies to get S(2, k).

3/21

Recursive construction of Sierpi´ nski graph S(n, k)

• 1. Start with the complete graph: S(1, k) = Kk.
• 2. Copy it k times.
• 3. Add one edge between each pair of copies to get S(2, k).
• 4. ”New” vertex i is vertex i of copy i.

1 2 3

3/21

Recursive construction of Sierpi´ nski graph S(n, k)

• 1. Start with the complete graph: S(1, k) = Kk.
• 2. Copy it k times.
• 3. Add one edge between each pair of copies to get S(2, k).
• 4. ”New” vertex i is vertex i of copy i.
• 5. Repeat to obtain S(3, k),S(4, k),...

1 2 3 1 2 3

3/21

Examples of Sierpi´ nski graphs

S(n, k) : k vertices in the complete graph, n iterations

• S(4,3)
• S(5,2)

4/21

Sierpi´ nski graphs: definition with words

Vertex set of Sierpi´ nski graphs: {1, . . . , k}n Edge between u1u2...un and v1v2...vn, if there is 1 ≤ j ≤ n s.t:

• ui = vi if i < j,
• uj = vj,
• ui = vj and vi = uj if i > j

u = v = w x y . . . y w y x . . . x Extreme vertex x: vertex x...x

5/21

Sierpi´ nski graphs: definition with words

111 112 113 122 123 121 132 131 133 211 212 213 232 311 313 321 322 223 221 233 231 331 323 312 332 222 333 u = v = w x y . . . y w y x . . . x

6/21

About Sierpi´ nski Graphs

• Introduced in 1997 by Klavˇ

zar and Milutinovi´ c

• S(n, 3) are Hano¨

ı graphs (n = number of disks):

7/21

About Sierpi´ nski Graphs

• Introduced in 1997 by Klavˇ

zar and Milutinovi´ c

• S(n, 3) are Hano¨

ı graphs (n = number of disks):

↔

7/21

About Sierpi´ nski Graphs

• Introduced in 1997 by Klavˇ

zar and Milutinovi´ c

• S(n, 3) are Hano¨

ı graphs (n = number of disks):

↔

7/21

About Sierpi´ nski Graphs

• Introduced in 1997 by Klavˇ

zar and Milutinovi´ c

• S(n, 3) are Hano¨

ı graphs (n = number of disks):

↔

• S(n, k): Hano¨

ı game on k rods and n disks with another move:

↔

7/21

Study of Sierpi´ nski Graphs

• Metric properties (shortest path, diameter. . . ), (Klavˇ

zar, Milutinovi´ c, 1997)

• Hamiltonicity, (Klavˇ

zar, Milutinovi´ c, 1997)

• Automorphism group, (Klavˇ

zar, Mohar, 2000)

• Crossing number, (Klavˇ

zar, Mohar, 2000)

• Perfect codes, identifying codes, (a, b)-codes,... (Klavˇ

zar et al. 2002, Beaudou et al. 2010,. . . )

• ...

8/21

Generalization

⇒

Complete graph Kk General graph G on k vertices S(n, k) S(n, G)

9/21

Recursive construction

1 4 2 3 5

→

1 4 2 3 5 1 4 2 3 5 1 4 2 3 5 1 4 2 3 5 1 4 2 3 5 copy 1 copy 2 copy 5 copy 4 copy 3

• A copy for each vertex of G

10/21

Recursive construction

1 4 2 3 5

→

1 4 2 3 5 1 4 2 3 5 1 4 2 3 5 1 4 2 3 5 1 4 2 3 5 copy 1 copy 2 copy 5 copy 4 copy 3

• A copy for each vertex of G
• Edge ij in G → edge between i in copy j and j in copy i

10/21

Recursive construction

1 4 2 3 5

→

1 4 2 3 5 1 4 2 3 5 1 4 2 3 5 1 4 2 3 5 1 4 2 3 5 copy 1 copy 2 copy 5 copy 4 copy 3

• A copy for each vertex of G
• Edge ij in G → edge between i in copy j and j in copy i

10/21

Recursive construction

1 4 2 3 5

→

1 2 3 4 5

• A copy for each vertex of G
• Edge ij in G → edge between i in copy j and j in copy i
• Extreme vertex ( ) : vertex i of copy i

10/21

Recursive construction

• 10/21

Equivalent definition with words

Vertex set: {1, . . . , k}n Edge between u1u2...un and v1v2...vn, if there is 1 ≤ j ≤ n s.t:

• ui = vi if i < j,
• uj = vj, and (uj, vj) ∈ E(G)
• ui = vj and vi = uj if i > j

u = v = w x y . . . y w y x . . . x (x, y) ∈ E(G) Extreme vertex x: vertex x...x

11/21

Basic properties

Idea: express property on S(n, G) with property of G

• Chromatic number: χ(S(n, G)) = χ(G).
• Distance between extreme vertex i and

extreme vertex j: (2n − 1) ∗ dG(i, j).

• Connexity: a cutting edge in G implies

some cutting edges in S(n, G).

• Self-similar graph: vertices starting with

u,|u| = i induce S(n − i, G).

12/21

Perfect codes in classical Sierpi´ nski graphs

There is always a perfect code in S(n, Kk) Theorem (Klavˇ

zar, Milutinovi´ c, Petr,2002)

Each vertex is dominated by exactly one code vertex

13/21

General case

If any packing of G leaves at least two vertices, S(n, G) does never have a perfect code. Proposition If G does not have a perfect code, there is an equivalence between:

• S(n, G) has a perfect code for one n > 1.
• S(n, G) has a perfect code for any n > 1.
• S(2, G) has a perfect code.
• G has a ’special’ 2-factor.

Proposition

14/21

Special case: Power of cycles

We consider G = C r

k:

k = i[2r + 1] r n = 1 n = 2 n > 3 i > 1 no no no i = 1 r even no no no i = 1 r odd no yes yes i = 0 1 < r < n

2

yes yes no i = 0 r = 1 or r ≥ n

2

yes yes yes

15/21

Elements of proof

We have k = 1[2r + 1]. Therefore:

• G = C r

k does not have a perfect code.

• There is a packing of G that let only one vertex not dominated.

We have to show that: S(2, G) has a perfect code ⇒ r is odd

16/21

Elements of proof

Assume there is a perfect code C in S(2, G), consider a copy of G: i . . . copy i

• C induces a packing in copy i of G,

17/21

Elements of proof

Assume there is a perfect code C in S(2, G), consider a copy of G: i . . . copy i c(i)

• C induces a packing in copy i of G,
• c(i) : bridge vertex in C

17/21

Elements of proof

Assume there is a perfect code C in S(2, G), consider a copy of G: i . . . copy i c(i) a(i)

• C induces a packing in copy i of G,
• c(i) : bridge vertex in C
• a(i) : bridge vertex not dominated inside copy i

17/21

Elements of proof

Assume there is a perfect code C in S(2, G), consider a copy of G: i . . . copy i c(i) a(i) c(a(i)) a(c(i))

• C induces a packing in copy i of G,
• c(i) : bridge vertex in C
• a(i) : bridge vertex not dominated inside copy i
• |a(i) − c(i)| = r + 1, a ◦ c = c ◦ a = id

17/21

Elements of proof

• For each i : a(i),c(i),
• |a(i) − c(i)| = r + 1, a ◦ c = c ◦ a = id

i a(i) c(i)

18/21

Elements of proof

• For each i : a(i),c(i),
• |a(i) − c(i)| = r + 1, a ◦ c = c ◦ a = id

i a(i) c(i)

• Construction of a 2-factor on {1, . . . , k}.

18/21

Elements of proof

• For each i : a(i),c(i),
• |a(i) − c(i)| = r + 1, a ◦ c = c ◦ a = id
• Construction of a 2-factor on {1, . . . , k}.
• Induces a mapping on {1, . . . , r + 1}, so r + 1 is even.

18/21

When G has a perfect code

k = i[2r + 1] r n = 1 n = 2 n > 3 i > 1 no no no i = 1 r even no no no i = 1 r odd no yes yes i = 0 1 < r < n

2

yes yes no i = 0 r = 1 or r ≥ n

2

yes yes yes We study weak perfect code: packing with only extreme vertices not dominated.

• Caracterized by the status of extreme vertices: only few possibilities.
• Weak perfect code for n give weak perfect code for n + 1.
• no weak perfect code for some n0 ⇒ no perfect code for n ≥ n0.

19/21

Result on the automorphism group

• Description and size:

id id

20/21

Result on the automorphism group

• Description and size:

id id

|Aut(n, C4)| = O(24n−2+...)

20/21

Result on the automorphism group

• Description and size:

|Aut(n, C4)| = O(24n−2+...)

• Distinguishing number

→ with the distinguishing number of G :

• 20/21

Result on the automorphism group

• Description and size:

|Aut(n, C4)| = O(24n−2+...)

• Distinguishing number

→ with the distinguishing number of G :

• D(S(n, G)) = max(max

x∈V D(G/x), 2)

20/21

Perspectives

• Hamiltonicity,
• Crossing number,
• Identifying codes, (a, b)-codes,
• Any local property,...

21/21

Perspectives

• Hamiltonicity,
• Crossing number,
• Identifying codes, (a, b)-codes,
• Any local property,...

Thank you for your attention !

21/21