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 20 th , 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 ) = K k . 3 1 2 3/21
Recursive construction of Sierpi´ nski graph S ( n , k ) 1. Start with the complete graph: S (1 , k ) = K k . 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 ) = K k . 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 ) = K k . 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 . 3 1 2 3/21
Recursive construction of Sierpi´ nski graph S ( n , k ) 1. Start with the complete graph: S (1 , k ) = K k . 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 ),... 3 3 1 2 1 2 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 nski graphs: { 1 , . . . , k } n Vertex set of Sierpi´ Edge between u 1 u 2 ... u n and v 1 v 2 ... v n , if there is 1 ≤ j ≤ n s.t: • u i = v i if i < j , • u j � = v j , • u i = v j and v i = u j if i > j y . . . y u = w x y v = w x . . . x Extreme vertex x: vertex x ... x 5/21
Sierpi´ nski graphs: definition with words 333 331 332 313 323 311 322 312 321 133 233 131 132 231 232 113 123 213 223 111 222 112 121 122 211 212 221 y . . . y u = w x y v = w 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
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