On identifying codes and Bondy’s theorem on “induced subsets” F. Foucaud 1 , E. Guerrini 2 , M. Kovše 1 , R. Naserasr 1 , A. Parreau 2 , P. Valicov 1 1: LaBRI, Université de Bordeaux, France 2: Institut Fourier, Université de Grenoble, France ANR IDEA (ANR-08-EMER-007, 2009-2011) 8FCC (LRI, Orsay) - July 02, 2010 F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 1 / 27
Outline 1 Introduction, definitions, examples 2 Finite and infinite undirected graphs 3 Finite digraphs 4 An application to Bondy’s theorem F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 2 / 27
Locating a fire in a building simple, undirected graph: models a building F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 3 / 27
Locating a fire in a building simple, undirected graph: models a building a c d b e f F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 4 / 27
Locating a fire in a building simple detectors: able to detect a fire in a neighbouring room { b , c } { b } { c } a c d { b , c } b e f { b } { b , c } F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 5 / 27
Locating a fire in a building simple detectors: able to detect a fire in a neighbouring room { b , c } { b } { c } a c d { b , c } b e f { b } { b , c } F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 6 / 27
Locating a fire in a building simple detectors: able to detect a fire in a neighbouring room { b , c , d } { a , b } { c , d } a c d { a , b , c } b e f { b } { b , c } F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 7 / 27
Identifying codes: definition Let N [ u ] be the set of vertices v s.t. d ( u , v ) ≤ 1 Definition: identifying code of a graph G (Karpovsky et al. 1998) subset C of V such that: C is a dominating set in G : for all u ∈ V , N [ u ] ∩ C � = ∅ , and C is a separating code in G : ∀ u � = v of V , N [ u ] ∩ C � = N [ v ] ∩ C F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 8 / 27
Identifying codes: definition Let N [ u ] be the set of vertices v s.t. d ( u , v ) ≤ 1 Definition: identifying code of a graph G (Karpovsky et al. 1998) subset C of V such that: C is a dominating set in G : for all u ∈ V , N [ u ] ∩ C � = ∅ , and C is a separating code in G : ∀ u � = v of V , N [ u ] ∩ C � = N [ v ] ∩ C Notation γ ID ( G ) : minimum cardinality of an identifying code of G F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 8 / 27
Identifiable graphs Remark: not all graphs have an identifying code u and v are twins if N [ u ] = N [ v ] . A graph is identifiable iff it is twin-free (i.e. it has no twin vertices). F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 9 / 27
Identifiable graphs Remark: not all graphs have an identifying code u and v are twins if N [ u ] = N [ v ] . A graph is identifiable iff it is twin-free (i.e. it has no twin vertices). Non-identifiable graphs F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 9 / 27
Identifiable graphs Remark: not all graphs have an identifying code u and v are twins if N [ u ] = N [ v ] . A graph is identifiable iff it is twin-free (i.e. it has no twin vertices). Non-identifiable graphs F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 9 / 27
An upper bound Theorem (Gravier, Moncel, 2007) Let G be a finite identifiable graph with n vertices and at least one edge. Then γ ID ( G ) ≤ n − 1. F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 10 / 27
An upper bound Theorem (Gravier, Moncel, 2007) Let G be a finite identifiable graph with n vertices and at least one edge. Then γ ID ( G ) ≤ n − 1. Corollary The only finite graphs having their whole vertex set as a minimum identifying code are the stable sets K n . F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 10 / 27
The graph A ∞ ( − 2 , 0 ) ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( 2 , 0 ) ... ... Infinite clique on Z ... ... Infinite clique on Z ( − 2 , 0 ) ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( 2 , 0 ) F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 11 / 27
The graph A ∞ ( − 2 , 0 ) ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( 2 , 0 ) ... ... Infinite clique on Z ... ... Infinite clique on Z ( − 2 , 0 ) ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( 2 , 0 ) Proposition (Charon, Hudry, Lobstein, 2007) A ∞ needs all its vertices in any identifying code. F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 11 / 27
The graph A ∞ ( − 2 , 0 ) ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( 2 , 0 ) ... ... Infinite clique on Z ... ... Infinite clique on Z ( − 2 , 0 ) ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( 2 , 0 ) Proposition (Charon, Hudry, Lobstein, 2007) A ∞ needs all its vertices in any identifying code. F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 12 / 27
The graph A ∞ ( − 2 , 0 ) ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( 2 , 0 ) ... ... Infinite clique on Z ... ... Infinite clique on Z ( − 2 , 0 ) ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( 2 , 0 ) Proposition (Charon, Hudry, Lobstein, 2007) A ∞ needs all its vertices in any identifying code. F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 13 / 27
Constructing infinite graphs Construction of Ψ( H , ρ ) H : finite or infinite simple graph with perfect matching ρ : perfect matching of H Replace every edge { u , v } of ρ by a copy of A ∞ complete join along the other edges of H x 2 Ψ y 2 − → y 1 y 3 x 1 x 3 H and ρ = { x 1 y 1 , x 2 y 2 , x 3 y 3 }
Constructing infinite graphs Construction of Ψ( H , ρ ) H : finite or infinite simple graph with perfect matching ρ : perfect matching of H Replace every edge { u , v } of ρ by a copy of A ∞ complete join along the other edges of H x 2 Ψ y 2 − → Y 1 y 1 y 3 X 1 x 1 x 3 A ∞ H and ρ = { x 1 y 1 , x 2 y 2 , x 3 y 3 }
Constructing infinite graphs Construction of Ψ( H , ρ ) H : finite or infinite simple graph with perfect matching ρ : perfect matching of H Replace every edge { u , v } of ρ by a copy of A ∞ complete join along the other edges of H X 2 A ∞ Y 2 x 2 Ψ y 2 − → Y 1 y 1 y 3 X 1 x 1 x 3 A ∞ H and ρ = { x 1 y 1 , x 2 y 2 , x 3 y 3 }
Constructing infinite graphs Construction of Ψ( H , ρ ) H : finite or infinite simple graph with perfect matching ρ : perfect matching of H Replace every edge { u , v } of ρ by a copy of A ∞ complete join along the other edges of H X 2 A ∞ Y 2 x 2 Ψ y 2 − → Y 1 y 1 y 3 X 1 x 1 x 3 A ∞ A ∞ H and ρ = { x 1 y 1 , x 2 y 2 , x 3 y 3 } Y 3 X 3
Constructing infinite graphs Construction of Ψ( H , ρ ) H : finite or infinite simple graph with perfect matching ρ : perfect matching of H Replace every edge { u , v } of ρ by a copy of A ∞ complete join along the other edges of H X 2 A ∞ Y 2 x 2 Ψ y 2 − → ⊳ ⊳ Y 1 ⊲ ⊲ y 1 y 3 X 1 x 1 x 3 A ∞ A ∞ H and ρ = { x 1 y 1 , x 2 y 2 , x 3 y 3 } Y 3 X 3 F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 14 / 27
The classification Theorem (F., Guerrini, Kovše, Naserasr, Parreau, Valicov, 2010) Let G be a connected infinite identifiable undirected graph. The only identifying code of G is V ( G ) if and only if G = Ψ( H , ρ ) for some graph H with a perfect matching ρ . F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 15 / 27
Idcodes in digraphs Let N − [ u ] be the set of incoming neighbours of u , plus u Definition: identifying code of a digraph D = ( V , A ) subset C of V such that: C is a dominating set in D : for all u ∈ V , N − [ u ] ∩ C � = ∅ , and C is a separating code in D : for all u � = v , N − [ u ] ∩ C � = N − [ v ] ∩ C { c , f } { b } { c } a c d { b , c , e } b e f { e } { b , c , f } F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 16 / 27
Idcodes in digraphs Let N − [ u ] be the set of incoming neighbours of u , plus u Definition: identifying code of a digraph D = ( V , A ) subset C of V such that: C is a dominating set in D : for all u ∈ V , N − [ u ] ∩ C � = ∅ , and C is a separating code in D : for all u � = v , N − [ u ] ∩ C � = N − [ v ] ∩ C { c , f } { b } { c } a c d { b , c , e } b e f { e } { b , c , f } Definition − → γ ID ( D ) : minimum size of an identifying code of D F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 16 / 27
Which graphs need n vertices? Two operations D 1 ⊕ D 2 : disjoint union of D 1 and D 2 − → ⊳ ( D ) : D joined to K 1 by incoming arcs only D 1 D 2 D − → D 1 ⊕ D 2 ⊳ ( D ) F. Foucaud (LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 17 / 27
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