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Watching Systems in Complete Bipartite Graphs C. Hernando M. Mora - - PowerPoint PPT Presentation

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Introduction Watching systems and watching number Complete bipartite graphs Summary Watching Systems in Complete Bipartite Graphs C. Hernando M. Mora I. M. Pelayo Depts. Matemtica Aplicada I, II, III Universitat Politcnica de Catalunya

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Introduction Watching systems and watching number Complete bipartite graphs Summary

Watching Systems in Complete Bipartite Graphs

  • C. Hernando
  • M. Mora
  • I. M. Pelayo
  • Depts. Matemàtica Aplicada I, II, III

Universitat Politècnica de Catalunya

VIII JMDA. Almería, 10-13 de Julio de 2012

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary

Outline

Introduction Detection devices and graphs Identifying codes Watching systems and watching number Watching systems Bounds of the watching number Complete bipartite graphs Bounds of the watching number Concrete values

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Detection devices and graphs Identifying codes

Detection devices

◮ Detection devices located at some vertices of a graph ◮ Detect and locate an object placed at any vertex of a graph ◮ Dominating/total dominating sets ◮ Locating sets

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Detection devices and graphs Identifying codes

Detection devices

◮ Detection devices located at some vertices of a graph ◮ Detect and locate an object placed at any vertex of a graph ◮ Dominating/total dominating sets ◮ Locating sets

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Detection devices and graphs Identifying codes

Detection devices and graphs

푢 푣 푢 푣 푢 푣 푢 푣 Dominating set Locating set Locating-dominating set Identifying set

12 21 22 11 13

푤 푤 푡

011 100 110 001

1100 0110 0010 1010 1101 0101 1001

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Detection devices and graphs Identifying codes

Definitions

G = (V, E) graph,

◮ N(u) = {v : uv ∈ E} ◮ N[u] = {u} ∪ N(u) ◮ twin vertices: N[u] = N[v] ◮ twin-free graph: it has no pair of twin vertices ◮ dominating set: S ⊆ V s.t. for all v ∈ V \ S, S ∩ N(v) = ∅ ◮ dominating number, γ(G): minimum size of a dominating

set of G

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Detection devices and graphs Identifying codes

Definitions

G = (V, E) graph,

◮ N(u) = {v : uv ∈ E} ◮ N[u] = {u} ∪ N(u) ◮ twin vertices: N[u] = N[v] ◮ twin-free graph: it has no pair of twin vertices ◮ dominating set: S ⊆ V s.t. for all v ∈ V \ S, S ∩ N(v) = ∅ ◮ dominating number, γ(G): minimum size of a dominating

set of G

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Detection devices and graphs Identifying codes

Definitions

G = (V, E) graph,

◮ N(u) = {v : uv ∈ E} ◮ N[u] = {u} ∪ N(u) ◮ twin vertices: N[u] = N[v] ◮ twin-free graph: it has no pair of twin vertices ◮ dominating set: S ⊆ V s.t. for all v ∈ V \ S, S ∩ N(v) = ∅ ◮ dominating number, γ(G): minimum size of a dominating

set of G

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Detection devices and graphs Identifying codes

Identifying codes [Karpovsky, Chakrabarty, Levitin, 1998]

Identifying code in a graph G = (V, E): S ⊆ V s.t. the sets N[v] ∩ C, v ∈ V(G), are all nonempty and distinct.

◮ label of vertex v: LC(v) = N[v] ∩ C ◮ identifying number, i(G): minimum size of an identifying

code of G

◮ Identifying codes exist only in twin-free graphs.

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Detection devices and graphs Identifying codes

Identifying codes [Karpovsky, Chakrabarty, Levitin, 1998]

Identifying code in a graph G = (V, E): S ⊆ V s.t. the sets N[v] ∩ C, v ∈ V(G), are all nonempty and distinct.

◮ label of vertex v: LC(v) = N[v] ∩ C ◮ identifying number, i(G): minimum size of an identifying

code of G

◮ Identifying codes exist only in twin-free graphs.

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Watching systems Bounds of the watching number

Watching systems

[Auger, Charon, Hudry, Lobstein, 2010]

Watching system in a graph G = (V, E) graph: W = {w1, w2, . . . , wk} where wi = (l(wi), A(wi)), with l(wi) = vi ∈ V(G) and A(wi) ⊆ N[vi], for all i ∈ {1, 2, . . . , k}, s.t. the sets LW(v) = {w ∈ W : v ∈ A(wi)} are all nonempty and distinct.

◮ wi is a watcher located at vertex l(wi) that checks its

watching zone, A(wi)

◮ LW(v) is the label of vertex v

Several watchers at the same vertex, each watcher checks its watching zone

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Watching systems Bounds of the watching number

Watching systems

[Auger, Charon, Hudry, Lobstein, 2010]

Watching system in a graph G = (V, E) graph: W = {w1, w2, . . . , wk} where wi = (l(wi), A(wi)), with l(wi) = vi ∈ V(G) and A(wi) ⊆ N[vi], for all i ∈ {1, 2, . . . , k}, s.t. the sets LW(v) = {w ∈ W : v ∈ A(wi)} are all nonempty and distinct.

◮ wi is a watcher located at vertex l(wi) that checks its

watching zone, A(wi)

◮ LW(v) is the label of vertex v

Several watchers at the same vertex, each watcher checks its watching zone

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Watching systems Bounds of the watching number

Watching systems

[Auger, Charon, Hudry, Lobstein, 2010]

Watching system in a graph G = (V, E) graph: W = {w1, w2, . . . , wk} where wi = (l(wi), A(wi)), with l(wi) = vi ∈ V(G) and A(wi) ⊆ N[vi], for all i ∈ {1, 2, . . . , k}, s.t. the sets LW(v) = {w ∈ W : v ∈ A(wi)} are all nonempty and distinct.

◮ wi is a watcher located at vertex l(wi) that checks its

watching zone, A(wi)

◮ LW(v) is the label of vertex v

Several watchers at the same vertex, each watcher checks its watching zone

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Watching systems Bounds of the watching number

Watching number

◮ watching number, w(G): minimum size of a watching

system of G

◮ minimum watching system: watching system of cardinality

w(G)

◮ Watching systems exist for all graphs ◮ w(G) ≤ i(G) if there exists at least an identifying code in G ◮ A watching system remais so if we add edges

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Watching systems Bounds of the watching number

Watching number

◮ watching number, w(G): minimum size of a watching

system of G

◮ minimum watching system: watching system of cardinality

w(G)

◮ Watching systems exist for all graphs ◮ w(G) ≤ i(G) if there exists at least an identifying code in G ◮ A watching system remais so if we add edges

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Watching systems Bounds of the watching number

Example

G = K1,6: i(G) = 6, w(G) = 3

W = {w1, w2, w3}, l(wi) = 7 A(w1) = {1, 4, 5, 7}, A(w2) = {2, 4, 6, 7}, A(w3) = {3, 5, 6, 7}

7 1 2 3 4 5 6 w1, w2, w3 w1 w2 w3 w1, w2 w1, w3 w2, w3 w1, w2, w3

LW(1) = {w1}, LW(2) = {w2}, LW(3) = {w3}, LW(4) = {w1, w2}, LW(5) = {w1, w3}, LW(6) = {w2, w3}, LW(7) = {w1, w2, w3}.

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Watching systems Bounds of the watching number

General bounds of the watching number

◮ w(G) ≥ ⌈log2(n + 1)⌉ ◮ Complete graphs, stars, graphs s.t. ∆ = n − 1 attain this

bound

◮ w(G) ≥ γ(G) ◮ w(G) ≤ γ(G) ⌈log2(∆ + 2)⌉ ◮ w(G) ≤ i(G) , if G is twin-free ◮ w(G) ≤ w(H) for any spanning subgraph H of G ◮ w(G) ≤ 2n 3 , if G is a connected graph of order 3 or ≥ 5

[Auger, Charon, Hudry, Lobstein, to appear]

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Watching systems Bounds of the watching number

General bounds of the watching number

◮ w(G) ≥ ⌈log2(n + 1)⌉ ◮ Complete graphs, stars, graphs s.t. ∆ = n − 1 attain this

bound

◮ w(G) ≥ γ(G) ◮ w(G) ≤ γ(G) ⌈log2(∆ + 2)⌉ ◮ w(G) ≤ i(G) , if G is twin-free ◮ w(G) ≤ w(H) for any spanning subgraph H of G ◮ w(G) ≤ 2n 3 , if G is a connected graph of order 3 or ≥ 5

[Auger, Charon, Hudry, Lobstein, to appear]

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Watching systems Bounds of the watching number

General bounds of the watching number

◮ w(G) ≥ ⌈log2(n + 1)⌉ ◮ Complete graphs, stars, graphs s.t. ∆ = n − 1 attain this

bound

◮ w(G) ≥ γ(G) ◮ w(G) ≤ γ(G) ⌈log2(∆ + 2)⌉ ◮ w(G) ≤ i(G) , if G is twin-free ◮ w(G) ≤ w(H) for any spanning subgraph H of G ◮ w(G) ≤ 2n 3 , if G is a connected graph of order 3 or ≥ 5

[Auger, Charon, Hudry, Lobstein, to appear]

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Watching systems Bounds of the watching number

General bounds of the watching number

◮ w(G) ≥ ⌈log2(n + 1)⌉ ◮ Complete graphs, stars, graphs s.t. ∆ = n − 1 attain this

bound

◮ w(G) ≥ γ(G) ◮ w(G) ≤ γ(G) ⌈log2(∆ + 2)⌉ ◮ w(G) ≤ i(G) , if G is twin-free ◮ w(G) ≤ w(H) for any spanning subgraph H of G ◮ w(G) ≤ 2n 3 , if G is a connected graph of order 3 or ≥ 5

[Auger, Charon, Hudry, Lobstein, to appear]

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Watching systems Bounds of the watching number

Watching number and identifying number

  • f some families

w(Pn) = n + 1 2

  • i(Pn) =

n + 1 2

  • w(Cn) =

3 , if n = 4; ⌈ n

2⌉

, otherwise. i(Cn) =    3, if n = 4, 5;

n 2,

if n ≥ 6 even;

n+3 2 ,

if n ≥ 7 odd.

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Complete bipartite graphs

Kr,s, 2 ≤ r ≤ s

◮ γ(Kr,s) = 2 ◮ i(Kr,s) = r + s − 2

W = {wi : i ∈ [m] } watching system in Kr,s

◮ V(Kr,s) = V1 ∪ V2, |V1| = r, |V2| = s ◮ L(W) = {l(wi) : i ∈ [m]} ⊆ V ◮ L1(W) = L(W) ∩ V1, L2(W) = L(W) ∩ V2,

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Complete bipartite graphs

Kr,s, 2 ≤ r ≤ s

◮ γ(Kr,s) = 2 ◮ i(Kr,s) = r + s − 2

W = {wi : i ∈ [m] } watching system in Kr,s

◮ V(Kr,s) = V1 ∪ V2, |V1| = r, |V2| = s ◮ L(W) = {l(wi) : i ∈ [m]} ⊆ V ◮ L1(W) = L(W) ∩ V1, L2(W) = L(W) ∩ V2,

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Bounds

w0(r, s) = ⌈log2(r + s + 1)⌉ Bounds:

◮ w0(r, s) ≤ w(Kr,s) ≤ ⌈log2 r⌉ + ⌈log2 s⌉

Both bounds are tight:

◮ w(K3,16) = w0(3, 16) = 5 ◮ w(K8,11) = ⌈log2 8⌉ + ⌈log2 11⌉ = 7

Particular case:

◮ w(K2,s) = w0(2, s) = ⌈log2(s + 3)⌉

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Bounds

w0(r, s) = ⌈log2(r + s + 1)⌉ Bounds:

◮ w0(r, s) ≤ w(Kr,s) ≤ ⌈log2 r⌉ + ⌈log2 s⌉

Both bounds are tight:

◮ w(K3,16) = w0(3, 16) = 5 ◮ w(K8,11) = ⌈log2 8⌉ + ⌈log2 11⌉ = 7

Particular case:

◮ w(K2,s) = w0(2, s) = ⌈log2(s + 3)⌉

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Bounds

w0(r, s) = ⌈log2(r + s + 1)⌉ Bounds:

◮ w0(r, s) ≤ w(Kr,s) ≤ ⌈log2 r⌉ + ⌈log2 s⌉

Both bounds are tight:

◮ w(K3,16) = w0(3, 16) = 5 ◮ w(K8,11) = ⌈log2 8⌉ + ⌈log2 11⌉ = 7

Particular case:

◮ w(K2,s) = w0(2, s) = ⌈log2(s + 3)⌉

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Bounds

w0(r, s) = ⌈log2(r + s + 1)⌉ Bounds:

◮ w0(r, s) ≤ w(Kr,s) ≤ ⌈log2 r⌉ + ⌈log2 s⌉

Both bounds are tight:

◮ w(K3,16) = w0(3, 16) = 5 ◮ w(K8,11) = ⌈log2 8⌉ + ⌈log2 11⌉ = 7

Particular case:

◮ w(K2,s) = w0(2, s) = ⌈log2(s + 3)⌉

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Watching Systems in Complete Bipartite Graphs

Consider Kr,s, 2 ≤ r ≤ s:

◮ If a watching system has 2 watchers at a same vertex, we

  • btain another watching system by placing one of them at

another vertex of the same stable set

◮ A watching system with all watchers located in the same

stable set has size at least max{r, ⌈log2(r + s + 1)⌉}

◮ A watching system with at least a watcher in each stable

set has size > w0(r, s)

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Watching Systems in Complete Bipartite Graphs

Consider Kr,s, 2 ≤ r ≤ s:

◮ If a watching system has 2 watchers at a same vertex, we

  • btain another watching system by placing one of them at

another vertex of the same stable set

◮ A watching system with all watchers located in the same

stable set has size at least max{r, ⌈log2(r + s + 1)⌉}

◮ A watching system with at least a watcher in each stable

set has size > w0(r, s)

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Watching Systems in Complete Bipartite Graphs

Consider Kr,s, 2 ≤ r ≤ s:

◮ If a watching system has 2 watchers at a same vertex, we

  • btain another watching system by placing one of them at

another vertex of the same stable set

◮ A watching system with all watchers located in the same

stable set has size at least max{r, ⌈log2(r + s + 1)⌉}

◮ A watching system with at least a watcher in each stable

set has size > w0(r, s)

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Attaining the lower bound

If 2 ≤ r ≤ s,

◮ If Kr,s = K5,5, w(Kr,s) = w0(r, s) if and only if r ≤ w0(r, s).

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Attaining the lower bound

If 2 ≤ r ≤ s,

◮ If Kr,s = K5,5, w(Kr,s) = w0(r, s) if and only if r ≤ w0(r, s).

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Not attaining the lower bound

If r > w0(r, s),

◮ There is a minimum watching system W satisfying

|L1(W)| ≥ |L2(W)|

◮ w(Kr,s) = min{m : m = h+k, r ≤ k +2h −1, s ≤ h+2k −1} ◮ If 6 ≤ r = s, then w(Kr,r) = w0(r, r) ◮ For each r ≥ 3, there is a minimum watching system of Kr,r

such that 0 ≤ |L1(W)| − |L2(W)| ≤ 1

◮ For each r ≥ 3, if nh = h + 2h,

w(Kr,r) =

  • 2h,

if nh−1 < r < nh for some h ≥ 2; 2h + 1, if r = nh for some h ≥ 2.

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Not attaining the lower bound

If r > w0(r, s),

◮ There is a minimum watching system W satisfying

|L1(W)| ≥ |L2(W)|

◮ w(Kr,s) = min{m : m = h+k, r ≤ k +2h −1, s ≤ h+2k −1} ◮ If 6 ≤ r = s, then w(Kr,r) = w0(r, r) ◮ For each r ≥ 3, there is a minimum watching system of Kr,r

such that 0 ≤ |L1(W)| − |L2(W)| ≤ 1

◮ For each r ≥ 3, if nh = h + 2h,

w(Kr,r) =

  • 2h,

if nh−1 < r < nh for some h ≥ 2; 2h + 1, if r = nh for some h ≥ 2.

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Feasible values

w(Kr,s) = w0(r, s), if r ≤ w0(r, s); w0(r, s) ≤ w(Kr,s) ≤ r, if r > w0(r, s). r ≤ w0 r > w0

r r w = w0 w = w0 = r w0 w0 w w = r

w0(r, s) ≤ w(Kr,s) ≤ max{r, w0(r, s)} Given a, b, c with 2 ≤ a ≤ b ≤ c, find r, s, such that 2 ≤ r ≤ s and w0(Kr,s) = a, w(Kr,s) = b, max{r, w0(r, s)} = c

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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SLIDE 36

Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Feasible values

w(Kr,s) = w0(r, s), if r ≤ w0(r, s); w0(r, s) ≤ w(Kr,s) ≤ r, if r > w0(r, s). r ≤ w0 r > w0

r r w = w0 w = w0 = r w0 w0 w w = r

w0(r, s) ≤ w(Kr,s) ≤ max{r, w0(r, s)} Given a, b, c with 2 ≤ a ≤ b ≤ c, find r, s, such that 2 ≤ r ≤ s and w0(Kr,s) = a, w(Kr,s) = b, max{r, w0(r, s)} = c

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Feasible values

w(Kr,s) = w0(r, s), if r ≤ w0(r, s); w0(r, s) ≤ w(Kr,s) ≤ r, if r > w0(r, s). r ≤ w0 r > w0

r r w = w0 w = w0 = r w0 w0 w w = r

w0(r, s) ≤ w(Kr,s) ≤ max{r, w0(r, s)} Given a, b, c with 2 ≤ a ≤ b ≤ c, find r, s, such that 2 ≤ r ≤ s and w0(Kr,s) = a, w(Kr,s) = b, max{r, w0(r, s)} = c

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Feasible values

Existence of r, s such that w0(Kr,s) = a, w(Kr,s) = b, and max{r, w0(r, s)} = c:

◮ If 2 ≤ a = b = c, a solution is r = a and s = 2a − a − 1 ◮ If 2 ≤ a = b < c, there is no solution ◮ If 2 ≤ a < b = c, there is solution if and only if

a ≥ log2(2c−3 + c + 3).

◮ If 2 ≤ a < b < c, if there is a solution, then

a + ⌈log2(c − a + 3)⌉ − 2 ≤ b ≤ a + ⌈log2(c − a + 1)⌉

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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Introduction Watching systems and watching number Complete bipartite graphs Summary Bounds of the watching number Concrete values

Watching number of Kr,s

w(K5,5) = 4, and for s ≥ r ≥ 3, not both equal to 5: w(Kr,s) = w0, if r ≤ w0; w(Kr,s) = w0 + 1, if r = w0 + 1; w(Kr,s) ∈ {w0 + 1, w0 + 2}, if r = w0 + 2; w(Kr,s) ∈ {w0 + ⌈log2(r − w0 + 1)⌉, w0 + ⌈log2(r − w0 + 2)⌉ − 1, w0 + ⌈log2(r − w0 + 3)⌉ − 2} if r ≥ w0 + 3. The identifying number of the complete bipartite graph Kr,s is r + s − 2 !

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

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SLIDE 40

Introduction Watching systems and watching number Complete bipartite graphs Summary

Summary

◮ Watching systems as an extension of identifying codes

◮ Watching systems exist in all graphs ◮ w(G) ≤ i(G) if G has at least an identifying code

◮ Watching systems and watching number of complete

bipartite graphs

◮ Open problems

◮ Watching number in bipartite graphs and other families ◮ Graphs with minimum watching number

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

slide-41
SLIDE 41

Introduction Watching systems and watching number Complete bipartite graphs Summary

Summary

◮ Watching systems as an extension of identifying codes

◮ Watching systems exist in all graphs ◮ w(G) ≤ i(G) if G has at least an identifying code

◮ Watching systems and watching number of complete

bipartite graphs

◮ Open problems

◮ Watching number in bipartite graphs and other families ◮ Graphs with minimum watching number

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs

slide-42
SLIDE 42

Introduction Watching systems and watching number Complete bipartite graphs Summary

Summary

◮ Watching systems as an extension of identifying codes

◮ Watching systems exist in all graphs ◮ w(G) ≤ i(G) if G has at least an identifying code

◮ Watching systems and watching number of complete

bipartite graphs

◮ Open problems

◮ Watching number in bipartite graphs and other families ◮ Graphs with minimum watching number

  • C. Hernando, M. Mora, I. M. Pelayo

Watching Systems in Complete Bipartite Graphs