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

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