SLIDE 1 On identifying codes and Bondy’s theorem
- n “induced subsets”
- F. Foucaud1, E. Guerrini2, M. Kovše1,
- R. Naserasr1, A. Parreau2, P. Valicov1
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
(LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 1 / 27
SLIDE 2 Outline
1 Introduction, definitions, examples 2 Finite and infinite undirected graphs 3 Finite digraphs 4 An application to Bondy’s theorem
(LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 2 / 27
SLIDE 3 Locating a fire in a building
simple, undirected graph: models a building
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SLIDE 4 Locating a fire in a building
simple, undirected graph: models a building
a b c d e f
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SLIDE 5 Locating a fire in a building
simple detectors: able to detect a fire in a neighbouring room
a b c d e f {b} {b, c} {b, c} {c} {b} {b, c}
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SLIDE 6 Locating a fire in a building
simple detectors: able to detect a fire in a neighbouring room
a b c d e f {b} {b, c} {b, c} {c} {b} {b, c}
(LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 6 / 27
SLIDE 7 Locating a fire in a building
simple detectors: able to detect a fire in a neighbouring room
a b c d e f {a, b} {a, b, c} {b, c, d} {c, d} {b} {b, c}
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SLIDE 8 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
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SLIDE 9 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
γID(G): minimum cardinality of an identifying code of G
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SLIDE 10 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).
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SLIDE 11 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
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SLIDE 12 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
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SLIDE 13 An upper bound
Theorem (Gravier, Moncel, 2007)
Let G be a finite identifiable graph with n vertices and at least one edge. Then γID(G) ≤ n − 1.
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SLIDE 14 An upper bound
Theorem (Gravier, Moncel, 2007)
Let G be a finite identifiable graph with n vertices and at least one edge. Then γID(G) ≤ n − 1.
Corollary
The only finite graphs having their whole vertex set as a minimum identifying code are the stable sets Kn.
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SLIDE 15 The graph A∞ ...
(−2, 0) (−1, 0) (0, 0) (1, 0) (2, 0)
... ...
(−2, 0) (−1, 0) (0, 0) (1, 0) (2, 0)
...
Infinite clique on Z Infinite clique on Z
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SLIDE 16 The graph A∞ ...
(−2, 0) (−1, 0) (0, 0) (1, 0) (2, 0)
... ...
(−2, 0) (−1, 0) (0, 0) (1, 0) (2, 0)
...
Infinite clique on Z Infinite clique on Z
Proposition (Charon, Hudry, Lobstein, 2007)
A∞ needs all its vertices in any identifying code.
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SLIDE 17 The graph A∞ ...
(−2, 0) (−1, 0) (0, 0) (1, 0) (2, 0)
... ...
(−2, 0) (−1, 0) (0, 0) (1, 0) (2, 0)
...
Infinite clique on Z Infinite clique on Z
Proposition (Charon, Hudry, Lobstein, 2007)
A∞ needs all its vertices in any identifying code.
(LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 12 / 27
SLIDE 18 The graph A∞ ...
(−2, 0) (−1, 0) (0, 0) (1, 0) (2, 0)
... ...
(−2, 0) (−1, 0) (0, 0) (1, 0) (2, 0)
...
Infinite clique on Z Infinite clique on Z
Proposition (Charon, Hudry, Lobstein, 2007)
A∞ needs all its vertices in any identifying code.
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SLIDE 19
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
x1 y1 y2 x2 y3 x3 H and ρ = {x1y1, x2y2, x3y3}
Ψ
− →
SLIDE 20
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
x1 y1 y2 x2 y3 x3 H and ρ = {x1y1, x2y2, x3y3}
Ψ
− →
A∞ Y1 X1
SLIDE 21
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
x1 y1 y2 x2 y3 x3 H and ρ = {x1y1, x2y2, x3y3}
Ψ
− →
A∞ Y1 X1 A∞ X2 Y2
SLIDE 22
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
x1 y1 y2 x2 y3 x3 H and ρ = {x1y1, x2y2, x3y3}
Ψ
− →
A∞ Y1 X1 A∞ X2 Y2 A∞ Y3 X3
SLIDE 23 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
x1 y1 y2 x2 y3 x3 H and ρ = {x1y1, x2y2, x3y3}
Ψ
− →
A∞ Y1 X1 A∞ X2 Y2 A∞ Y3 X3 ⊲ ⊳ ⊲ ⊳
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SLIDE 24 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.
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SLIDE 25 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
a b c d e f {b} {b, c, e} {c, f } {c} {e} {b, c, f }
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SLIDE 26 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
a b c d e f {b} {b, c, e} {c, f } {c} {e} {b, c, f }
Definition
− → γID(D): minimum size of an identifying code of D
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SLIDE 27 Which graphs need n vertices?
Two operations
D1 ⊕ D2: disjoint union of D1 and D2 − → ⊳ (D): D joined to K1 by incoming arcs only
D1 D2 D1 ⊕ D2 D − → ⊳ (D)
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SLIDE 28 Which graphs need n vertices?
Two operations
D1 ⊕ D2: disjoint union of D1 and D2 − → ⊳ (D): D joined to K1 by incoming arcs only
Definition
Let (K1, ⊕, − → ⊳ ) be the digraphs which can be built from K1 by successive applications of ⊕ and − → ⊳ , starting with K1.
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SLIDE 29
Which graphs need n vertices?
Two operations
D1 ⊕ D2: disjoint union of D1 and D2 − → ⊳ (D): D joined to K1 by incoming arcs only
Definition
Let (K1, ⊕, − → ⊳ ) be the digraphs which can be built from K1 by successive applications of ⊕ and − → ⊳ , starting with K1.
SLIDE 30
Which graphs need n vertices?
Two operations
D1 ⊕ D2: disjoint union of D1 and D2 − → ⊳ (D): D joined to K1 by incoming arcs only
Definition
Let (K1, ⊕, − → ⊳ ) be the digraphs which can be built from K1 by successive applications of ⊕ and − → ⊳ , starting with K1.
SLIDE 31
Which graphs need n vertices?
Two operations
D1 ⊕ D2: disjoint union of D1 and D2 − → ⊳ (D): D joined to K1 by incoming arcs only
Definition
Let (K1, ⊕, − → ⊳ ) be the digraphs which can be built from K1 by successive applications of ⊕ and − → ⊳ , starting with K1.
SLIDE 32
Which graphs need n vertices?
Two operations
D1 ⊕ D2: disjoint union of D1 and D2 − → ⊳ (D): D joined to K1 by incoming arcs only
Definition
Let (K1, ⊕, − → ⊳ ) be the digraphs which can be built from K1 by successive applications of ⊕ and − → ⊳ , starting with K1.
SLIDE 33 Which graphs need n vertices?
Two operations
D1 ⊕ D2: disjoint union of D1 and D2 − → ⊳ (D): D joined to K1 by incoming arcs only
Definition
Let (K1, ⊕, − → ⊳ ) be the digraphs which can be built from K1 by successive applications of ⊕ and − → ⊳ , starting with K1.
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SLIDE 34 A characterization
Proposition
Let D be a digraph of (K1, ⊕, − → ⊳ ) on n vertices. − → γID(D) = n.
D1 D2 D1 ⊕ D2 D − → ⊳ (D)
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SLIDE 35 A characterization
Theorem (F., Naserasr, Parreau, 2010)
Let D be an identifiable digraph on n vertices. − → γID(G) = n iff D ∈ (K1, ⊕, − → ⊳ ).
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SLIDE 36 A characterization
Theorem (F., Naserasr, Parreau, 2010)
Let D be an identifiable digraph on n vertices. − → γID(G) = n iff D ∈ (K1, ⊕, − → ⊳ ).
A useful proposition
Let D be a digraph with − → γID(G) = n − 1, then there is a vertex x of D such that − → γID(D − x) = n − 1
Proof of the Theorem
By contradiction: take a minimum counterexample, D By the proposition, there is a vertex x such that − → γID(D − x) = |V (D − x)| − 1. Hence D − x ∈ (K1, ⊕, − → ⊳ ). Show that by adding a vertex to D − x, we either stay in the family or decrease − → γID.
(LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 20 / 27
SLIDE 37 A theorem of Bondy
Theorem on “induced subsets” (Bondy, 1972)
Let S = {S1, S2, · · · Sn} 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 S1 − X ′, S2 − X ′, · · · Sn − X ′ are all distinct.
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SLIDE 38 A theorem of Bondy
Theorem on “induced subsets” (Bondy, 1972)
Let S = {S1, S2, · · · Sn} 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 S1 − X ′, S2 − X ′, · · · Sn − X ′ are all distinct.
Example with k = 0
X = {1,2,3,4} and S = {{1,4}, {3}, {2,4}, {1,2,4}}
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SLIDE 39 A theorem of Bondy
Theorem on “induced subsets” (Bondy, 1972)
Let S = {S1, S2, · · · Sn} 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 S1 − X ′, S2 − X ′, · · · Sn − 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}}
(LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 21 / 27
SLIDE 40 A theorem of Bondy
Theorem on “induced subsets” (Bondy, 1972)
Let S = {S1, S2, · · · Sn} 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 S1 − X ′, S2 − X ′, · · · Sn − 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}}
The result is best possible
X = {1, 2, 3, 4} and S = {∅, {1}, {2}, {3}, {4}}
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SLIDE 41 Bipartite representation
Bipartite representation
We can build a bipartite graph B = (S + X, E) where Si connected to x iff x ∈ Si. Bondy’s theorem states that there exists a code C ⊆ X which separates S 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 {1} {1, 3} {2, 3} {1, 3, 4} X 1 2 3 4
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SLIDE 42 Bipartite separating 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 code separating S with X in B is a separating code of D.
Example
X = {1, 2, 3, 4} and S = {{1}, {1, 3}, {2, 3}, {1, 3, 4}}
S {1} {1, 3} {2, 3} {1, 3, 4} X 1 2 3 4
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SLIDE 43 Bipartite separating 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 code separating S with X in B is a separating code of D.
Example
X = {1, 2, 3, 4} and S = {{1}, {1, 3}, {2, 3}, {1, 3, 4}}
S {1} {1, 3} {2, 3} {1, 3, 4} X 1 2 3 4 {1}/1 {2, 3}/2 {1, 3}/3 {1, 3, 4}/4
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SLIDE 44 Application to Bondy’s setting
Corollary (F., Naserasr, Parreau, 2010)
In Bondy’s theorem, if for any good choice of x we have Si − x = ∅ for some Si, then B is the neighbourhood graph of a digraph in (K1, ⊕, − → ⊳ ).
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SLIDE 45 Application to Bondy’s setting
Corollary (F., Naserasr, Parreau, 2010)
In Bondy’s theorem, if for any good choice of x we have Si − x = ∅ for some Si, then B is the neighbourhood graph of a digraph in (K1, ⊕, − → ⊳ ).
Proof (1)
If |X| > |S| (|X| = n + k, k > 0): by Bondy’s theorem we can remove k + 1 ≥ 2 elements of X. At most one of them can create ∅, so we choose another one!
(LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 24 / 27
SLIDE 46 Application to Bondy’s setting
Corollary (F., Naserasr, Parreau, 2010)
In Bondy’s theorem, if for any good choice of x we have Si − x = ∅ for some Si, then B is the neighbourhood graph of a digraph in (K1, ⊕, − → ⊳ ).
Proof (2)
If |X| = |S| If B has a perfect matching: use our theorem. Otherwise, by Hall’s theorem, there is a subset X1 of X s.t. |X1| > |N(X1)|.
S X X0 N(X0) X1 N(X1)
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SLIDE 47 Application to Bondy’s setting
Corollary (F., Naserasr, Parreau, 2010)
In Bondy’s theorem, if for any good choice of x we have Si − x = ∅ for some Si, then B is the neighbourhood graph of a digraph in (K1, ⊕, − → ⊳ ).
Proof (2)
If |X| = |S| If B has a perfect matching: use our theorem. Otherwise, by Hall’s theorem, there is a subset X1 of X s.t. |X1| > |N(X1)|.
S X X0 N(X0) N(X1) X1
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SLIDE 48 Conclusion
Future work
Classify infinite digraphs D with V (D) as their only identifying code What about graphs having V (D) − x as an only identifying code?
(LaBRI, U. Bordeaux) On id. codes and related probems 8FCC - 02/07/2010 27 / 27
