Bounds on the size of identifying codes for graphs of maximum degree - PowerPoint PPT Presentation

Bounds on the size of identifying codes for graphs of maximum degree ∆ Florent Foucaud joint work with Ralf Klasing, Adrian Kosowski, André Raspaud Université Bordeaux 1 September 2009 F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 1 / 25

Locating a fire in a building simple, undirected graph : models a building a c d b e f F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 2 / 25

Locating a fire in a building simple detectors : able to detect a fire in a neighbouring room goal : locate an eventual fire { b , c } { b } { c } a c d { b , c } b b c a • e f b • • { b } { b , c } • • c d • e • f • • F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 3 / 25

Locating a fire in a building simple detectors : able to detect a fire in a neighbouring room goal : locate an eventual fire fire in room f { b , c } { b } { c } a c d { b , c } b b c a • e f b • • { b } { b , c } • • c d • e • f • • F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 4 / 25

Locating a fire in a building simple detectors : able to detect a fire in a neighbouring room goal : locate an eventual fire fire in room f the identifying sets of all vertices must be distinct { b , c , d } { a , b } { c , d } a c d { a , b , c } b a b c d a • • e f b • • • { b } { b , c } c • • • d • • e • f • • F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 5 / 25

Identifying codes : definition Definition : identifying code of a graph G = ( V , E ) (Karpovsky et al. 1998 [2]) subset C of V such that : C is a dominating set in G , and for all distinct u , v of V , u and v have distinct identifying sets : N [ u ] ∩ C � = N [ v ] ∩ C F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 6 / 25

Identifying codes : definition Definition : identifying code of a graph G = ( V , E ) (Karpovsky et al. 1998 [2]) subset C of V such that : C is a dominating set in G , and for all distinct u , v of V , u and v have distinct identifying sets : N [ u ] ∩ C � = N [ v ] ∩ C Remark Note : close to locating-dominating sets (Slater, Rall 84 [4]) F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 6 / 25

Identifying codes : definition Definition : identifying code of a graph G = ( V , E ) (Karpovsky et al. 1998 [2]) subset C of V such that : C is a dominating set in G , and for all distinct u , v of V , u and v have distinct identifying sets : N [ u ] ∩ C � = N [ v ] ∩ C Remark Note : close to locating-dominating sets (Slater, Rall 84 [4]) Notation γ id ( G ) : minimum cardinality of an identifying code in a graph G F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 6 / 25

Identifiable graphs Remark : not all graphs admit an identifying code u and v are twin vertices if N [ u ] = N [ v ] . A graph is identifiable iff it has no twin vertices. F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 7 / 25

Identifiable graphs Remark : not all graphs admit an identifying code u and v are twin vertices if N [ u ] = N [ v ] . A graph is identifiable iff it has no twin vertices. Non-identifiable graphs F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 7 / 25

Lower bound and maximum degree Thm (Karpovski et al. 98 [2]) Let G be an identifiable graph with n vertices. Then γ id ( G ) ≥ ⌈ log 2 ( n + 1 ) ⌉ . F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 8 / 25

Lower bound and maximum degree Thm (Karpovski et al. 98 [2]) Let G be an identifiable graph with n vertices. Then γ id ( G ) ≥ ⌈ log 2 ( n + 1 ) ⌉ . Characterization The graphs reaching this bound have been characterized (Moncel 06 [3]) F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 8 / 25

Lower bound and maximum degree Thm (Karpovski et al. 98 [2]) Let G be an identifiable graph with n vertices. Then γ id ( G ) ≥ ⌈ log 2 ( n + 1 ) ⌉ . Characterization The graphs reaching this bound have been characterized (Moncel 06 [3]) Thm (Karpovski et al. 98 [2]) Let G be an identifiable graph with n vertices and maximum degree ∆ . 2 n Then γ id ( G ) ≥ ∆ + 2. F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 8 / 25

Graphs reaching the lower bound Characterization n vertices 2 n independent set C of size ∆+ 2 (id. code) every vertex of C has exactly ∆ neighbours ∆ n ∆+ 2 vertices connected to exactly 2 code vertices each F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 9 / 25

Graphs reaching the lower bound - example Example : D =Petersen graph, ∆ = 3, n = 10 F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 10 / 25

A general upper bound Thm (Gravier, Moncel 07 [1]) Let G be an identifiable connected graph with n ≥ 3 vertices. Then γ id ( G ) ≤ n − 1. F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 13 / 25

A general upper bound Thm (Gravier, Moncel 07 [1]) Let G be an identifiable connected graph with n ≥ 3 vertices. Then γ id ( G ) ≤ n − 1. Thm (Gravier, Moncel 07 [1]) For all n ≥ 3, there exist identifiable graphs with n vertices with γ id ( G ) = n − 1. F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 13 / 25

Upper bound - example Example : the star K 1 , n − 1 F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 14 / 25

Upper bound and maximum degree Remark All these graphs have a high maximum degree ∆( G ) : n − 1 or n − 2. F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 15 / 25

Result - general case Thm (F., Klasing, Kosowski and Raspaud 09) Let G be a connected identifiable graph of maximum degree ∆ . n Then γ id ( G ) ≤ n − Θ(∆ 4 ) . n If G is regular, γ id ( G ) ≤ n − Θ(∆ 2 ) . F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 16 / 25

Result - general case Thm (F., Klasing, Kosowski and Raspaud 09) Let G be a connected identifiable graph of maximum degree ∆ . n Then γ id ( G ) ≤ n − Θ(∆ 4 ) . n If G is regular, γ id ( G ) ≤ n − Θ(∆ 2 ) . Sketch of the proof Greedily construct a 4-independant (resp. 2-independent) set S : distance between two vertices is at least 5 (resp. 3) take C = V \ S as a code C must be modified locally F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 16 / 25

Connected cliques Take any ∆ -regular graph H with m vertices replace any vertex of H by a clique of ∆ vertices F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 17 / 25

Connected cliques Take any ∆ -regular graph H with m vertices replace any vertex of H by a clique of ∆ vertices Example : H = K 4 F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 17 / 25

Connected cliques Take any ∆ -regular graph H with m vertices Replace any vertex of H by a clique of ∆ vertices Exemple : H = K 4 F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 18 / 25

Connected cliques Take any ∆ -regular graph H with m vertices replace any vertex of H by a clique of ∆ vertices Exemple : H = K 4 For every clique, at least ∆ − 1 vertices in the code ⇒ γ id ( G ) ≥ m · (∆ − 1 ) = n − n ∆ F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 19 / 25

Large codes in triangle-free graphs Proposition Let K m , m be the complete bipartite graph with n = 2 m vertices. id ( K m , m ) = 2 m − 2 = n − n ∆ . F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 20 / 25

Large codes in triangle-free graphs Proposition Let K m , m be the complete bipartite graph with n = 2 m vertices. id ( K m , m ) = 2 m − 2 = n − n ∆ . Thm (Bertrand et al. 05) Let T h k be the k -ary tree with h levels and n vertices. k 2 n � � n id ( T h k ) = = n − . k 2 + k + 1 ∆ − 1 + 1 ∆ F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 20 / 25

Triangle-free graphs - Result Thm (F., Klasing, Kosowski and Raspaud 09) Let G be a connected triangle-free identifiable graph G with n ≥ 3 vertices and maximum degree ∆ . n Then γ id ( G ) ≤ n − 3 ∆+ 3 . n If G is regular, γ id ( G ) ≤ n − 2 ∆+ 2 . F. Foucaud (U. Bordeaux 1) Bounds on id codes September 2009 21 / 25

Bounds on the size of identifying codes for graphs of maximum degree - PowerPoint PPT Presentation

Bounds on the size of identifying codes for graphs of maximum degree Florent Foucaud joint work with Ralf Klasing, Adrian Kosowski, Andr Raspaud Universit Bordeaux 1 September 2009 F. Foucaud (U. Bordeaux 1) Bounds on id codes

Semidefinite programming bounds for codes and anticodes in Cayley graphs Frank Vallentin

On Bounds for Batch Codes Jens Zumbr agel Institute of Algebra TU Dresden with Vitaly

Identifying codes and VC-dimension Aline Parreau University of Li` ege, Belgium Joint work

On the Exact Lower Bounds of Encoding Circuit Sizes of Hamming codes and Hadamard codes Zhengrui

Improved bounds on the word error probability of RA(2) codes with linear programming based

Neighbour-transitive codes in Johnson graphs Mark Ioppolo Centre for Mathematics of Symmetry and

15-853:Algorithms in the Real World Expander Graphs LDPC (Expander) codes 15-853

Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013,

Kernel-size lower bounds: accompanying exercises Andrew Drucker April 19, 2013 These

Spa$ally Coupled Turbo-Like Codes: Convolu$onal Codes on Graphs

An Analysis of Graph Cut Size for Transductive Learning Steve Hanneke Machine Learning

Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013,

Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013,

Codes on Random Geometric Graphs Dejan Vukobratovi Associate Professor, DEET-UNS University of

Information Theory Lecture 6 Block Codes and Finite Fields Codes: Roth (R) 12, 4.14

has been the focus of much interest recently, involving an interplay of methods Date : November 22,

Semidefinite programming bounds for codes D. Gijswijt 1 A. Schrijver 2 H. Tanaka 3 1 Department of

New upper bounds for mixed binary/ternary codes Bart Litjens Korteweg-de Vries Institute for

Universal lower bounds for the energy of spherical codes: lifting the Levenshtein framework P.

Locally identifying colorings of graphs Aline Parreau Joint work with: Louis Esperet, Sylvain

New upper bounds for nonbinary codes based on semidefinite programming and parity Sven Polak

Bounds on achievable rates of sparse quantum codes over the quantum erasure channel Nicolas

Lower Bounds on the Probability of Error of Polar Codes Boaz Shuval and Ido Tal Andrew and Erna

On identifying codes and Bondys theorem on induced subsets F. Foucaud 1 , E. Guerrini 2 ,