Probl` emes didentification dans les graphes Aline Parreau S - PowerPoint PPT Presentation

Identifying code in line graphs L ( G ) G L 16/42

Identifying code in line graphs L ( G ) G L 16/42

Identifying code in line graphs L ( G ) G L 16/42

Identifying code in line graphs L ( G ) G L 16/42

Identifying code in line graphs L ( G ) G L Identifying code 16/42

Identifying code in line graphs L ( G ) G L Edge identifying code Identifying code 16/42

Identifying code in line graphs L ( G ) G L Edge identifying code Identifying code γ EID ( G ) = γ ID ( L ( G )) Pendant edges Twins 16/42

Still difficult Edge-IDCode : Given G pendant-free and k , γ EID ( G ) ≤ k ? Theorem Foucaud, Gravier, Naserasr, P., Valicov, 2012 Edge-IDCode is NP-complete even for planar subcubic bipar- tite graphs with large girth. Reduction from Planar ( ≤ 3 , 3) -SAT . 17/42

Still difficult Edge-IDCode : Given G pendant-free and k , γ EID ( G ) ≤ k ? Theorem Foucaud, Gravier, Naserasr, P., Valicov, 2012 Edge-IDCode is NP-complete even for planar subcubic bipar- tite graphs with large girth. Reduction from Planar ( ≤ 3 , 3) -SAT . Corollary Identifying Code is NP-complete even for perfect planar 3- colorable line graphs with maximum degree 4. 17/42

Bounds using the number of vertices Proposition Foucaud, Gravier, Naserasr, P., Valicov, 2012 1 2 | V ( G ) | ≤ γ EID ( G ) ≤ 2 | V ( G ) | − 3 18/42

Bounds using the number of vertices Proposition Foucaud, Gravier, Naserasr, P., Valicov, 2012 1 2 | V ( G ) | ≤ γ EID ( G ) ≤ 2 | V ( G ) | − 3 • Lower Bound: a code must cover ≃ half of vertices. → Tight for hypercubes. 18/42

Bounds using the number of vertices Proposition Foucaud, Gravier, Naserasr, P., Valicov, 2012 1 2 | V ( G ) | ≤ γ EID ( G ) ≤ 2 | V ( G ) | − 3 • Lower Bound: a code must cover ≃ half of vertices. → Tight for hypercubes. • Upper Bound: a minimal code is 2-degenerate. → Tight only for K 4 . 18/42

Bounds using the number of vertices Proposition Foucaud, Gravier, Naserasr, P., Valicov, 2012 1 2 | V ( G ) | ≤ γ EID ( G ) ≤ 2 | V ( G ) | − 3 • Lower Bound: a code must cover ≃ half of vertices. → Tight for hypercubes. • Upper Bound: a minimal code is 2-degenerate. → Tight only for K 4 . → Infinite family with γ EID ( G ) = 2 | V ( G ) | − 6: · · · 18/42

Bounds using the number of vertices Proposition Foucaud, Gravier, Naserasr, P., Valicov, 2012 1 2 | V ( G ) | ≤ γ EID ( G ) ≤ 2 | V ( G ) | − 3 Corollary Edge-IDCode has a polynomial 4-approximation. • Best polynomial approximation for identifying codes in log( | V | ). (Laifenbeld, Trachtenberg, Berger-Wolf, 2006 and Gravier, Klasing, Moncel, 2008) 18/42

Bounds using the number of edges Proposition Foucaud, Gravier, Naserasr, P., Valicov, 2012 3 � | E ( G ) | ≤ γ EID ( G ) ≤ | E ( G ) | − 1 √ 2 2 • Upper Bound: from identifying code • Lower Bound: using the lower bound for vertices → Tight for: · · · � � � 19/42

Bounds using the number of edges Proposition Foucaud, Gravier, Naserasr, P., Valicov, 2012 3 � | E ( G ) | ≤ γ EID ( G ) ≤ | E ( G ) | − 1 √ 2 2 Corollary � If G is a line graph, γ ID ( G ) ≥ Θ( | V | ) 19/42

Conclusion for line graphs • Class of graph for which γ ID ( G ) ≥ Θ( � | V | ) (instead of Θ(log( | V | ))) . • Defined by forbidden induced subgraphs: • Is the lower bound still true with less restrictions? For other classes defined by forbidden induced subgraphs? → False for claw-free graphs. → True for interval graphs. 20/42

Part III A variation of identifying code: Identifying colorings of graphs 21/42

Some variations • Locating-dominating codes • Resolving sets • ( r , ≤ ℓ )-identifying codes • Weak and light codes • Tolerant identifying codes • Watching systems • Discriminating codes • Adaptative identifying codes • Locating colorings • ... 22/42

Some variations • Locating-dominating codes • Resolving sets • ( r , ≤ ℓ )-identifying codes • Weak and light codes • Tolerant identifying codes One more: • Watching systems Identifying coloring • Discriminating codes • Adaptative identifying codes • Locating colorings • ... 22/42

Proper coloring of graphs → Two adjacent vertices have different colors. χ ( G ) = 3 Chromatic number χ ( G ) : minimum number of colors needed 23/42

Proper coloring of graphs - a lower bound Clique number ω ( G ) : max. number of vertices that induces a complete graph ω ( G ) = 3 For any graph G , χ ( G ) ≥ ω ( G ) 24/42

Proper coloring of graphs - a lower bound Clique number ω ( G ) : max. number of vertices that induces a complete graph ω ( G ) = 4 For any graph G , χ ( G ) ≥ ω ( G ) 24/42

Proper coloring of graphs - a lower bound Clique number ω ( G ) : max. number of vertices that induces a complete graph ω ( G ) = 4 For any graph G , χ ( G ) ≥ ω ( G ) 24/42

...that is not always reached χ ( C 5 ) = 3 but ω ( C 5 ) = 2 25/42

...that is not always reached 2 1 1 2 3 χ ( C 5 ) = 3 but ω ( C 5 ) = 2 25/42

Perfect graphs Perfect graph (1963): G is perfect if ω ( H ) = χ ( H ) for any induced subgraph H of G Theorem Strong Perfect Graph Theorem (Chudnovsky et al. 2002) G is perfect if and only if it has no induced odd cycle or comple- ment of odd cycle with more than 4 vertices 26/42

A part of the big family of perfect graphs Perfect Line of bipartite Permutation Cograph Chordal Bipartite Split Interval k -trees Trees 27/42

Identification with colors Identifying codes Proper graph colorings 28/42

Identification with colors Identifying codes Proper graph colorings Identifying colorings 28/42

Locally identifying coloring • Proper vertex coloring c : V → N • local identification by the colors in the neighborhood: c ( N [ x ]) { 1 , 2 } { 1 , 2 , 3 } { 1 , 2 } 1 2 1 2 3 2 { 1 , 2 , 3 } { 2 , 3 } { 1 , 2 , 3 } c ( N [ x ]) � = c ( N [ y ]) for xy ∈ E • χ lid ( G ): min. number of colors in a lid-coloring of G . 29/42

An example: the path 30/42

An example: the path 1 2 3 4 1 2 3 4 30/42

An example: the path 1 2 3 4 1 2 3 4 1 , 2 1 , 2 , 3 2 , 3 , 4 1 , 3 , 4 1 , 2 , 4 1 , 2 , 3 2 , 3 , 4 3 , 4 30/42

An example: the path 1 2 3 4 1 2 3 4 1 , 2 1 , 2 , 3 2 , 3 , 4 1 , 3 , 4 1 , 2 , 4 1 , 2 , 3 2 , 3 , 4 3 , 4 χ lid ( P k ) ≤ 4 30/42

An example: the path 1 2 3 4 1 2 3 4 1 , 2 1 , 2 , 3 2 , 3 , 4 1 , 3 , 4 1 , 2 , 4 1 , 2 , 3 2 , 3 , 4 3 , 4 χ lid ( P k ) ≤ 4 With 3 colors : 1 2 3 2 1 2 3 2 1 , 2 1 , 2 , 3 2 , 3 1 , 2 , 3 1 , 2 1 , 2 , 3 2 , 3 2 , 3 χ lid ( P k ) = 3 iff k is odd. 30/42

Link with chromatic number • A lid-coloring is a proper coloring: χ lid ≥ χ . • No upper bound with χ . → complete graph K k subdivided twice: χ lid = k , χ = 3 31/42

Link with chromatic number • A lid-coloring is a proper coloring: χ lid ≥ χ . • No upper bound with χ . → complete graph K k subdivided twice: χ lid = k , χ = 3 1 1 31/42

Link with chromatic number • A lid-coloring is a proper coloring: χ lid ≥ χ . • No upper bound with χ . → complete graph K k subdivided twice: χ lid = k , χ = 3 1 3 2 1 31/42

Link with chromatic number • A lid-coloring is a proper coloring: χ lid ≥ χ . • No upper bound with χ . → complete graph K k subdivided twice: χ lid = k , χ = 3 1 3 2 1 • Not monotone: χ lid ( P 5 ) ≤ χ lid ( P 4 ) 31/42

χ lid is not monotone at all 32/42

χ lid is not monotone at all u χ lid ( G ) = 5 ≪ k = χ lid ( G − u ) 32/42

Study in perfect graphs Perfect L (bipartite) Permutation Cograph Chordal Bipartite Split Interval k -tree Tree 33/42

Study in perfect graphs Perfect L (bipartite) Permutation Cograph Chordal Bipartite ? Bipartite Split Interval k -tree Tree Tree 33/42

Bipartite graphs: the path 1 2 3 4 1 2 3 4 1 , 2 1 , 2 , 3 2 , 3 , 4 1 , 3 , 4 1 , 2 , 4 1 , 2 , 3 2 , 3 , 4 3 , 4 χ lid ( P k ) ≤ 4 34/42

Bipartite graphs are 4-lid-colorable L 0 L 1 L 2 L 3 L 4 35/42

Bipartite graphs are 4-lid-colorable → L 0 1 { 1 , 2 } → L 1 { 1 , 2 , 3 } 2 → { 2 , 3 , 4 } or { 2 , 3 } L 2 3 → L 3 4 { 1 , 3 , 4 } or { 3 , 4 } → L 4 { 1 , 4 } 1 35/42

Bipartite graphs are 4-lid-colorable → L 0 1 { 1 , 2 } → L 1 { 1 , 2 , 3 } 2 → { 2 , 3 , 4 } or { 2 , 3 } L 2 3 → L 3 4 { 1 , 3 , 4 } or { 3 , 4 } → L 4 { 1 , 4 } 1 If G is bipartite, χ lid ( G ) ≤ 4. 35/42

Bipartite graphs General bounds: 3 ≤ χ lid ( B ) ≤ 4. 36/42