On some problems in graph theory: from colourings to vertex identication

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1 On some problems in graph theory: from colourings to vertex identication F. Foucaud 1 Joint works with: E. Guerrini 1,2, R. Klasing 1, A. Kosowski 1,3, M. Kov²e 1,2, R. Naserasr 1, A. Parreau 2, A. Raspaud 1, P. Valicov 1 1: LaBRI, Bordeaux University, France 2: Institut Fourier, Grenoble University, France 3: Gda«sk University of Technology, Poland Calicut University, Kerala, India - September 4th, 2010 F. Foucaud (LaBRI) On some graph theory problems / 54

2 Outline 1 Graph theory - an introduction (denitions, colouring maps, the 4CT) [board] 2 Networks as graphs, the domination problem [board] 3 Complexity of graph theory problems - the P versus NP problem [board] 4 Identifying codes - denitions, complexity, extremal problems [slides] F. Foucaud (LaBRI) On some graph theory problems / 54

3 Locating an intruder in a building simple, undirected graph: models a building F. Foucaud (LaBRI) On some graph theory problems / 54

4 Locating an intruder in a building simple, undirected graph: models a building a c d b e f F. Foucaud (LaBRI) On some graph theory problems / 54

5 Locating an intruder in a building simple detectors: able to detect an intruder in a neighbouring room goal: locate an eventual intruder {b} {b, c} {c} a {b, c} c d b e f {b} {b, c} b c a b c d e f F. Foucaud (LaBRI) On some graph theory problems / 54

6 Locating an intruder in a building simple detectors: able to detect an intruder in a neighbouring room goal: locate an eventual intruder intruder in room f {b} {b, c} {c} a {b, c} c d b e f {b} {b, c} b c a b c d e f F. Foucaud (LaBRI) On some graph theory problems / 54

7 Locating an intruder in a building simple detectors: able to detect an intruder in a neighbouring room goal: locate an eventual intruder intruder in room f the identifying sets of all vertices must be distinct {a, b} {b, c, d} {c, d} a {a, b, c} c d b e f {b} {b, c} a b c d a b c d e f F. Foucaud (LaBRI) On some graph theory problems / 54

8 Identifying codes: denition Let N[u] be the set of vertices v s.t. d(u, v) 1. Denition: identifying code of 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 set in G : u v of V, N[u] C N[v] C F. Foucaud (LaBRI) On some graph theory problems / 54

9 Identifying codes: denition Let N[u] be the set of vertices v s.t. d(u, v) 1. Denition: identifying code of 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 set 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) On some graph theory problems / 54

10 Identiable graphs Remark: not all graphs have an identifying code u and v are twins if N[u] = N[v]. A graph is identiable i it is twin-free (i.e. it has no twin vertices). F. Foucaud (LaBRI) On some graph theory problems / 54

11 Identiable graphs Remark: not all graphs have an identifying code u and v are twins if N[u] = N[v]. A graph is identiable i it is twin-free (i.e. it has no twin vertices). Non-identiable graphs F. Foucaud (LaBRI) On some graph theory problems / 54

12 Identiable graphs Remark: not all graphs have an identifying code u and v are twins if N[u] = N[v]. A graph is identiable i it is twin-free (i.e. it has no twin vertices). Non-identiable graphs F. Foucaud (LaBRI) On some graph theory problems / 54

13 Hardness Decision problem ID-CODE Input: identiable graph G and an integer k Question: Is γ ID (G) k? Thm (Cohen et al. 01, Auger et al. 09) ID-CODE is NP-complete even in bipartite and planar graphs. F. Foucaud (LaBRI) On some graph theory problems / 54

14 Hardness Decision problem ID-CODE Input: identiable graph G and an integer k Question: Is γ ID (G) k? Thm (Cohen et al. 01, Auger et al. 09) ID-CODE is NP-complete even in bipartite and planar graphs. Optimization problem MIN ID-CODE Input: identiable graph G Output: γ ID (G) Thm (Trachtenberg et al. 06, Suomela 07, Gravier et al. 08) MIN ID-CODE can be approximated within a logarithmic factor, but not within a constant factor. F. Foucaud (LaBRI) On some graph theory problems / 54

15 An upper bound Thm (Gravier, Moncel 2007) Let G be a twin-free graph with n 3 vertices and at least one edge. Then γ ID (G) n 1. F. Foucaud (LaBRI) On some graph theory problems / 54

16 An upper bound Thm (Gravier, Moncel 2007) Let G be a twin-free graph with n 3 vertices and at least one edge. Then γ ID (G) n 1. Thm (Charon, Hudry, Lobstein, Skaggs, 2007) For all n 3, there exist twin-free graphs with n vertices and γ ID (G) = n 1. F. Foucaud (LaBRI) On some graph theory problems / 54

17 Upper bound - small examples Recall the denition C is a dominating set in G : for all u V, N[u] C, and C is a separating set in G : u v of V, N[u] C N[v] C F. Foucaud (LaBRI) On some graph theory problems / 54

18 Upper bound - stars Recall the denition C is a dominating set in G : for all u V, N[u] C, and C is a separating set in G : u v of V, N[u] C N[v] C F. Foucaud (LaBRI) On some graph theory problems / 54

19 Upper bound - complete graph minus max. matching Recall the denition C is a dominating set in G : for all u V, N[u] C, and C is a separating set in G : u v of V, N[u] C N[v] C F. Foucaud (LaBRI) On some graph theory problems / 54

20 A conjecture Conjecture (Charon, Hudry, Lobstein, 2008) γ ID (G) = n 1 i G {P 4, K n \ M, K 1,n 1}. F. Foucaud (LaBRI) On some graph theory problems / 54

21 A class of graphs called A G r : graph where x, y are adjacent i d(x, y) r in G Denition: graph A k V (A k ) = {x 1,..., x 2k }. x i connected to x j i j i k 1 A 1 = K 2 ; for k 2, A k = P k 1 2k x k+1 x k+2 x k+3... x 2k 1 x 2k Clique on {x k+1,..., x 2k } x 1 x 2 x 3... x k 1 x k Clique on {x 1,..., x k } F. Foucaud (LaBRI) On some graph theory problems / 54

22 A class of graphs called A - examples A 1 = K 2 A 2 = P 4 A 3 = P 2 6 A 4 = P 3 8 F. Foucaud (LaBRI) On some graph theory problems / 54

23 Properties Proposition Let k 2, n = 2k. γ ID (A k ) = n 1. Remark In every minimum code C of A k, there exists a vertex x such that C = N[x]. F. Foucaud (LaBRI) On some graph theory problems / 54

24 (A, ) Join operation G 1 G 2 : disjoint copies of G 1 and G 2 + all possible edges between G 1 and G 2 Denition Let (A, ) be the closure of graphs of A with respect to. F. Foucaud (LaBRI) On some graph theory problems / 54

25 (A, ) Join operation G 1 G 2 : disjoint copies of G 1 and G 2 + all possible edges between G 1 and G 2 Denition Let (A, ) be the closure of graphs of A with respect to. Proposition Let G be a graph of (A, ) \ {K 2 } with n vertices. γ ID (G) = n 1. F. Foucaud (LaBRI) On some graph theory problems / 54

26 (A, ) K 1 Proposition Let G be a graph of (A, ) with n 1 vertices. γ ID (G K 1 ) = n 1. F. Foucaud (LaBRI) On some graph theory problems / 54

27 A characterization Thm (F., Guerrini, Kov²e, Naserasr, Parreau, Valicov, 2010) Let G be a twin-free graph on n vertices. γ ID (G) = n 1 i G S (A, ) (A, ) K 1 and G K 2. F. Foucaud (LaBRI) On some graph theory problems / 54

28 A useful Proposition Proposition Let G be a twin-free graph and S V such that G S is twin-free. Then γ ID (G) γ ID (G S) + S. Corollary Let G be a graph with γ ID (G) = V (G) 1, then there is a vertex x of G such that γ ID (G x) = V (G x) 1 F. Foucaud (LaBRI) On some graph theory problems / 54

29 Proof ideas Proof By contradiction: take a minimum counterexample, G By the proposition, there is a vertex x such that γ ID (G x) = V (G x) 1. Hence G x S (A, ) (A, ) K 1 and G K 2. For the three cases, show that by adding a vertex to G x, we either stay in the family or decrease γ ID. F. Foucaud (LaBRI) On some graph theory problems / 54

30 What about r-id codes? - Denition Denition: r -identifying codes subset C of V such that: C is an r-dominating set in G : for all u V, B r (u) C, and C is an r-separating set in G : u v of V, B r (u) C B r (v) C F. Foucaud (LaBRI) On some graph theory problems / 54

31 What about r-id codes? - Denition Denition: r -identifying codes subset C of V such that: C is an r-dominating set in G : for all u V, B r (u) C, and C is an r-separating set in G : u v of V, B r (u) C B r (v) C Remark C is an r-identifying code of G i C is a 1-identifying code of G r γ ID r (G) = γ ID (G r ) {G γ ID r (G) = n 1} = {G G r (S (A, ) (A, ) K 1 )\K 2 } F. Foucaud (LaBRI) On some graph theory problems / 54

32 What about r-id codes? - Roots of A k Question Let r 2 and k 2. What are the graphs G such that G r = A k = P k 1 2k? F. Foucaud (LaBRI) On some graph theory problems / 54

33 What about r-id codes? - Roots of A k Question Let r 2 and k 2. What are the graphs G such that G r = A k = P k 1 2k? Partial answer All P s 2k such that s r = k 1. Example: (P 2 10 )2 = P 4 10 hence γ ID 2 (P 2 10 ) = 9. Those graphs minus some edges... And other ones! Example: G such that G 2 = P 4 10 (so γid(g) = 9) F. Foucaud (LaBRI) On some graph theory problems / 54

34 Idcodes in digraphs Let N [u] be the set of incoming neighbours of u, plus u Denition: 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 for all u v of V, N [u] C N [v] C F. Foucaud (LaBRI) On some graph theory problems / 54

35 Idcodes in digraphs Let N [u] be the set of incoming neighbours of u, plus u Denition: 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 for all u v of V, N [u] C N [v] C a c d b e f F. Foucaud (LaBRI) On some graph theory problems / 54

36 Idcodes in digraphs Let N [u] be the set of incoming neighbours of u, plus u Denition: 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 for all u v of V, N [u] C N [v] C a c d b e f Denition Let γ ID (D) be the minimum size of an identifying code of D F. Foucaud (LaBRI) On some graph theory problems / 54

37 Which graphs need n vertices? Two operations D 1 D 2 : disjoint union of D 1 and D 2 (D): D joined to K1 by incoming arcs only F. Foucaud (LaBRI) On some graph theory problems / 54

38 Which graphs need n vertices? Two operations D 1 D 2 : disjoint union of D 1 and D 2 (D): D joined to K1 by incoming arcs only Denition Let (K 1,, ) be the digraphs which can be built from K 1 by successive application of and. F. Foucaud (LaBRI) On some graph theory problems / 54

39 Which graphs need n vertices? Two operations D 1 D 2 : disjoint union of D 1 and D 2 (D): D joined to K1 by incoming arcs only Denition Let (K 1,, ) be the digraphs which can be built from K 1 by successive application of and. Remark Every element of (K 1,, ) is the transitive closure of a forest of rooted oriented trees. F. Foucaud (LaBRI) On some graph theory problems / 54

40 Which graphs need n vertices? Two operations D 1 D 2 : disjoint union of D 1 and D 2 (D): D joined to K1 by incoming arcs only Denition Let (K 1,, ) be the digraphs which can be built from K 1 by successive application of and. Remark Every element of (K 1,, ) is the transitive closure of a forest of rooted oriented trees. Proposition Let D be a digraph of (K 1,, ) on n vertices. γ ID (D) = n. F. Foucaud (LaBRI) On some graph theory problems / 54

41 A characterization Thm (F., Naserasr, 2010) Let D be a twin-free digraph on n vertices. γ ID (G) = n i D (K 1,, ). F. Foucaud (LaBRI) On some graph theory problems / 54

42 A characterization Thm (F., Naserasr, 2010) Let D be a twin-free digraph on n vertices. γ ID (G) = n i D (K 1,, ). A useful proposition (digraphs) Let D be a digraph with γ ID (G) = V (D) 1, then there is a vertex x of D such that γ ID (D x) = V (D x) 1 Proof By contradiction: take a minimum counterexample, D By the proposition, there is a vertex x such that γ ID (D x) = V (D x) 1. Hence D x (K 1,, ). Show that by adding a vertex to D x, we either stay in the family or decrease γ ID. F. Foucaud (LaBRI) On some graph theory problems / 54

43 A theorem of Bondy Theorem on induced subsets (Bondy, 1972) Let S = {S 1, S 2, S n } be a collection of distinct (possibly empty) subsets of an n + k-set X (k 0). Then there is a (k + 1)-subset X of X such that S 1 X, S 2 X, S n X are all distinct. F. Foucaud (LaBRI) On some graph theory problems / 54

44 A theorem of Bondy Theorem on induced subsets (Bondy, 1972) Let S = {S 1, S 2, S n } be a collection of distinct (possibly empty) subsets of an n + k-set X (k 0). Then there is a (k + 1)-subset X of X such that S 1 X, S 2 X, S n X are all distinct. Example with k = 0 X = {1, 2, 3, 4} and S = {{1, 4}, {3}, {2, 4}, {1, 2, 4}} F. Foucaud (LaBRI) On some graph theory problems / 54

45 A theorem of Bondy Theorem on induced subsets (Bondy, 1972) Let S = {S 1, S 2, S n } be a collection of distinct (possibly empty) subsets of an n + k-set X (k 0). Then there is a (k + 1)-subset X of X such that S 1 X, S 2 X, S n X are all distinct. Example with k = 0 X = {1, 2, 3, 4} and S = {{1, 4}, {3}, {2, 4}, {1, 2, 4}} Example with k = 1 X = {1, 2, 3, 4, 5} and S = {{1, 4, 5}, {3}, {2, 4, 5}, {1, 2, 4, 5}} F. Foucaud (LaBRI) On some graph theory problems / 54

46 A theorem of Bondy Theorem on induced subsets (Bondy, 1972) Let S = {S 1, S 2, S n } be a collection of distinct (possibly empty) subsets of an n + k-set X (k 0). Then there is a (k + 1)-subset X of X such that S 1 X, S 2 X, S n X are all distinct. Example with k = 0 X = {1, 2, 3, 4} and S = {{1, 4}, {3}, {2, 4}, {1, 2, 4}} Example with k = 1 X = {1, 2, 3, 4, 5} and S = {{1, 4, 5}, {3}, {2, 4, 5}, {1, 2, 4, 5}} False for k = 1 X = {1, 2, 3, 4} and S = {, {1}, {2}, {3}, {4}} F. Foucaud (LaBRI) On some graph theory problems / 54

47 A theorem of Bondy - proof Proof Note: if S 1, S 2 X and S 1 x = S 2 x, then S 1 S 2 = {x}. By contradiction: Construct a graph H = (S, E) where for each x X, choose one unique (i, j) s.t. S i S j = {x}, and connect S i to S j. Claim: H has no cycle - a contradiction! F. Foucaud (LaBRI) On some graph theory problems / 54

48 Saying the same thing in another language Bipartite representation We can build a bipartite graph B = (S + X, E) where S i connected to x i x S i. Bondy's theorem states that there exists a discriminating code (see Charon, Cohen, Lobstein, Hudry, 2006) of S using X of size at most X 1 in B. F. Foucaud (LaBRI) On some graph theory problems / 54

49 Saying the same thing in another language Bipartite representation We can build a bipartite graph B = (S + X, E) where S i connected to x i x S i. Bondy's theorem states that there exists a discriminating code (see Charon, Cohen, Lobstein, Hudry, 2006) of S using X of size at most X 1 in B. Example X = {1, 2, 3, 4} and S = {{1}, {1, 3}, {2, 3}, {1, 3, 4}} S X {1} {1, 3} {2, 3} {1, 3, 4} F. Foucaud (LaBRI) On some graph theory problems / 54

50 Discriminating codes and identifying codes Remark Let B be the bipartite graph representing S, X If B has a matching from S to X, B is the neighbourhood graph of a digraph D. A discriminating code in B is a separating set of D! F. Foucaud (LaBRI) On some graph theory problems / 54

51 Discriminating codes and identifying codes Remark Let B be the bipartite graph representing S, X If B has a matching from S to X, B is the neighbourhood graph of a digraph D. A discriminating code in B is a separating set of D! Example X = {1, 2, 3, 4} and S = {{1}, {1, 3}, {2, 3}, {1, 3, 4}} S X {1} {1, 3} {2, 3} {1, 3, 4} F. Foucaud (LaBRI) On some graph theory problems / 54

52 Discriminating codes and identifying codes Remark Let B be the bipartite graph representing S, X If B has a matching from S to X, B is the neighbourhood graph of a digraph D. A discriminating code in B is a separating set of D! Example X = {1, 2, 3, 4} and S = {{1}, {1, 3}, {2, 3}, {1, 3, 4}} S {1} {1, 3} {2, 3} X {1, 3}/3 {1, 3, 4}/4 {1, 3, 4} 4 {1}/1 {2, 3}/2 F. Foucaud (LaBRI) On some graph theory problems / 54

53 Application to Bondy's setting Corollary (F., Naserasr, 2010) In Bondy's theorem, if we have S i x = for some S i and for any good choice of x, then B is the neighbourhood graph of a digraph in (K 1,, ). In other words This happens i for every S i, S j S, S i S j S i S j or S j S i. F. Foucaud (LaBRI) On some graph theory problems / 54

54 Application to Bondy's setting - Hall's theorem Marriage theorem (Hall, 1935) Let B = (X + Y, E) be a bipartite graph. B has a matching from X to Y i for all X X, X N(X ). F. Foucaud (LaBRI) On some graph theory problems / 54

55 Application to Bondy's setting - a proof Proof (1) If X > S ( X = n + k, k 0): by Bondy's theorem we can remove k + 1 elements of X. At most one can create an, so we choose another one of the k + 1. ( ) By our theorem: γ ID = n separating set of size n 1 F. Foucaud (LaBRI) On some graph theory problems / 54

56 Application to Bondy's setting - a proof Proof (2) ( ) If B has a perfect matching: use our theorem. Otherwise, by Hall's theorem, there is a subset S 1 of S s.t. S 1 > N(S 1 ). S X S 1 N(S 1 ) F. Foucaud (LaBRI) On some graph theory problems / 54

57 Upper bound and : a corollary and a question Corollary Let G be a twin-free graph on n vertices and maximum degree n 3. Then γ ID (G) n 2. F. Foucaud (LaBRI) On some graph theory problems / 54

58 Upper bound and : a corollary and a question Corollary Let G be a twin-free graph on n vertices and maximum degree n 3. Then γ ID (G) n 2. Question Is γ ID bounded by a function of n and? F. Foucaud (LaBRI) On some graph theory problems / 54

59 Upper bound and - two propositions Proposition 1 Let G be a twin-free graph, and x a vertex of G. There exists a vertex y, d(x, y) 1, and V y is an identifying code of G. F. Foucaud (LaBRI) On some graph theory problems / 54

60 Upper bound and - two propositions Proposition 1 Let G be a twin-free graph, and x a vertex of G. There exists a vertex y, d(x, y) 1, and V y is an identifying code of G. Proposition 2 Let G be a twin-free graph, and I a 4-independent set of G (all distances 4). If for all x I, V x is an identifying code of G, V I is also one. F. Foucaud (LaBRI) On some graph theory problems / 54

61 A rst bound Corollary (F., Klasing, Kosowski, Raspaud, 2009) Let G be a twin-free graph of maximum degree. γ ID (G) n n Θ( 5 ). F. Foucaud (LaBRI) On some graph theory problems / 54

62 A rst bound Corollary (F., Klasing, Kosowski, Raspaud, 2009) Let G be a twin-free graph of maximum degree. γ ID (G) n Proof Consider a maximal 6-independant set I : distance between two vertices is at least 6 and I n Θ( 5 ) For every x I, let f (x) be the vertex found in Prop. 1. n Θ( 5 ). V f (I ) is an identifying code of size at most n I by Prop. 2. F. Foucaud (LaBRI) On some graph theory problems / 54

63 A rst bound Corollary (F., Klasing, Kosowski, Raspaud, 2009) Let G be a twin-free graph of maximum degree. γ ID (G) n Proof Consider a maximal 6-independant set I : distance between two vertices is at least 6 and I n Θ( 5 ) For every x I, let f (x) be the vertex found in Prop. 1. Remark n Θ( 5 ). V f (I ) is an identifying code of size at most n I by Prop. 2. Can be improved to n n Θ( 4 ) with a bit of modications. F. Foucaud (LaBRI) On some graph theory problems / 54

64 A rst bound Corollary (F., Klasing, Kosowski, Raspaud, 2009) Let G be a twin-free graph of maximum degree. γ ID (G) n Proof Consider a maximal 6-independant set I : distance between two vertices is at least 6 and I n Θ( 5 ) For every x I, let f (x) be the vertex found in Prop. 1. Remark n Θ( 5 ). V f (I ) is an identifying code of size at most n I by Prop. 2. Can be improved to n Question Is this bound sharp? n Θ( 4 ) with a bit of modications. F. Foucaud (LaBRI) On some graph theory problems / 54

65 Connected cliques Take any -regular graph H with m vertices replace any vertex of H by a clique of vertices F. Foucaud (LaBRI) On some graph theory problems / 54

66 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 (LaBRI) On some graph theory problems / 54

67 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 (LaBRI) On some graph theory problems / 54

68 Connected cliques Take any -regular graph H with m vertices replace any vertex of H by a clique of vertices Example: H = K 4 For every clique, at least 1 vertices in the code γ ID (G) = m ( 1) = n n F. Foucaud (LaBRI) On some graph theory problems / 54

69 Triangle-free graphs Thm (F., Klasing, Kosowski, Raspaud, 2009) Let G be a twin-free connected triangle-free graph G with n 3 vertices and maximum degree. Then γ ID (G) n n. If G has minimum 3 +3 degree 3, γ ID (G) n n F. Foucaud (LaBRI) On some graph theory problems / 54

70 Triangle-free graphs Thm (F., Klasing, Kosowski, Raspaud, 2009) Let G be a twin-free connected triangle-free graph G with n 3 vertices and maximum degree. Then γ ID (G) n n. If G has minimum 3 +3 degree 3, γ ID (G) n n Proof Consider a maximal independent set I : S C = V \I Some vertices may not be identied correctly n +1 modify C locally. It is possible to add not too much vertices to C. F. Foucaud (LaBRI) On some graph theory problems / 54

71 Is this bound sharp? Proposition Let K m,m be the complete bipartite graph with n = 2m vertices. γ ID (K m,m ) = 2m 2 = n n. F. Foucaud (LaBRI) On some graph theory problems / 54

72 Is this bound sharp? Proposition Let K m,m be the complete bipartite graph with n = 2m vertices. γ ID (K m,m ) = 2m 2 = n n. Thm (Bertrand et al. 05) Let T h be the k-ary tree with h levels and n k vertices.γ ID (T h) = k 2 n n = n. k k 2 + k F. Foucaud (LaBRI) On some graph theory problems / 54

73 A conjecture Conjecture (F., Klasing, Kosowski, Raspaud, 2009) Let G be a connected twin-free graph of maximum degree. Then γ ID (G) n n. +1 F. Foucaud (LaBRI) On some graph theory problems / 54

74 Graphs of girth at least 5 Thm (F., Klasing, Kosowski, Raspaud, 2009) Let G be a twin-free graph with n vertices, of minimum degree δ 2 and girth g 5. Then γ ID (G) 7n F. Foucaud (LaBRI) On some graph theory problems / 54

75 Graphs of girth at least 5 Thm (F., Klasing, Kosowski, Raspaud, 2009) Let G be a twin-free graph with n vertices, of minimum degree δ 2 and girth g 5. Then γ ID (G) 7n Proof Construct a DFS spanning tree T of G Partition the vertices into 4 classes V 0, V 1, V 2, V 3 depending on their level in T Take C = V \V i as a code, V i n 4 : V i 3n 4 C must be modied locally; the size of C might increase F. Foucaud (LaBRI) On some graph theory problems / 54

76 Graphs of girth at least 5 level 0 level 1 level 2 level 3 level 4 level 5 level 6 F. Foucaud (LaBRI) On some graph theory problems / 54

77 Graphs of girth at least 5 - bad example G k,p : δ = 2, = p + 2, n = (5p + 1)k C k (k = 9)... p times γ ID (G k,p ) = 3pk = 3 5 (n k) 3n 5 F. Foucaud (LaBRI) On some graph theory problems / 54

78 A general lower bound Thm (Karpovsky et al. 98) Let G be a twin-free graph with n vertices. Then γ ID (G) log 2 (n + 1). F. Foucaud (LaBRI) On some graph theory problems / 54

79 A general lower bound Thm (Karpovsky et al. 98) Let G be a twin-free graph with n vertices. Then γ ID (G) log 2 (n + 1). Characterization The graphs reaching this bound have been characterized (Moncel 06) F. Foucaud (LaBRI) On some graph theory problems / 54

80 Lower bound and maximum degree Thm (Karpovsky et al. 98) Let G be a twin-free graph with n vertices and maximum degree. Then γ ID (G) 2n + 2. F. Foucaud (LaBRI) On some graph theory problems / 54

81 Graphs reaching the lower bound Characterization (F., Klasing, Kosowski, Raspaud, 2009) n vertices independent set C of size 2n (id. code) +2 every vertex of C has exactly neighbours n +2 vertices connected to exactly 2 code vertices each F. Foucaud (LaBRI) On some graph theory problems / 54

82 Graphs reaching the lower bound Characterization (F., Klasing, Kosowski, Raspaud, 2009) n vertices independent set C of size 2n (id. code) +2 every vertex of C has exactly neighbours n +2 Construction vertices connected to exactly 2 code vertices each Take a simple -regular graph D (code) Put a new vertex on each edge of D Eventually add edges between the new vertices F. Foucaud (LaBRI) On some graph theory problems / 54

83 Graphs reaching the lower bound - example Example: D=Petersen graph, = 3, n = 10 F. Foucaud (LaBRI) On some graph theory problems / 54

84 Graphs reaching the lower bound - example Example: D=Petersen graph, = 3, n = 10 F. Foucaud (LaBRI) On some graph theory problems / 54

85 Graphs reaching the lower bound - example Example: D=Petersen graph, = 3, n = 10 F. Foucaud (LaBRI) On some graph theory problems / 54