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Cops and a rober Guarding a subgraph Single intruder versus single guard
Guarding a subgraph as a tool in pursuit-evasiongames
Drago Bokal,Gordana Radic
Department of mathematics and computer scienceFaculty of natural sciences and mathematics
University of Maribor, Slovenia
CANADAM, MontrealMay 2009
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
1 Cops and a roberIntroductionA single cop versus a single robberSome results
2 Guarding a subgraphGame description
3 Single intruder versus single guardNotationShadow function and intruding functionShadow and retractions
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Basic rules of the game
played on a connected graph G = (V , E )2 players: cop, robber
perfect information
cops move, robber moves
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Basic rules of the game
played on a connected graph G = (V , E )2 players: cop, robber
perfect information
cops move, robber moves
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Basic rules of the game
played on a connected graph G = (V , E )2 players: cop, robber
perfect information
cops move, robber moves
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Basic rules of the game
played on a connected graph G = (V , E )2 players: cop, robber
perfect information
cops move, robber moves
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Basic rules of the game
played on a connected graph G = (V , E )2 players: cop, robber
perfect information
cops move, robber moves
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Basic rules of the game
played on a connected graph G = (V , E )2 players: cop, robber
perfect information
cops move, robber moves
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Basic rules of the game
played on a connected graph G = (V , E )2 players: cop, robber
perfect information
cops move, robber moves
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Basic rules of the game
played on a connected graph G = (V , E )2 players: cop, robber
perfect information
cops move, robber moves
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
A single cop versus a single robber
cop-number: c(G )
G is cop-win ⇔ c(G ) = 1
Kn; n ∈N is cop-win
tree is cop-win
Cn; n ≥ 4 is robber-win
R. Nowakowski, P. Winkler (1984):
G is cop-win ⇔ G has elimination order of vertices.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
A single cop versus a single robber
cop-number: c(G )
G is cop-win ⇔ c(G ) = 1
Kn; n ∈N is cop-win
tree is cop-win
Cn; n ≥ 4 is robber-win
R. Nowakowski, P. Winkler (1984):
G is cop-win ⇔ G has elimination order of vertices.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
A single cop versus a single robber
cop-number: c(G )
G is cop-win ⇔ c(G ) = 1
Kn; n ∈N is cop-win
tree is cop-win
Cn; n ≥ 4 is robber-win
R. Nowakowski, P. Winkler (1984):
G is cop-win ⇔ G has elimination order of vertices.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
A single cop versus a single robber
cop-number: c(G )
G is cop-win ⇔ c(G ) = 1
Kn; n ∈N is cop-win
tree is cop-win
Cn; n ≥ 4 is robber-win
R. Nowakowski, P. Winkler (1984):
G is cop-win ⇔ G has elimination order of vertices.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Pursuit-evasion results I.
M. Aigner, M. Fromme (1984):G planar ⇒ c(G ) ≤ 3
A. Quilliot (1985):genus of G : γ(G ) = k⇒ c(G ) ≤ 3 + 2k
P. Frankl (1987):girth of G : g(G ) ≥ 8t − 3⇒ c(G ) > (δ(G )− 1)t
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Pursuit-evasion results I.
M. Aigner, M. Fromme (1984):G planar ⇒ c(G ) ≤ 3
A. Quilliot (1985):genus of G : γ(G ) = k⇒ c(G ) ≤ 3 + 2k
P. Frankl (1987):girth of G : g(G ) ≥ 8t − 3⇒ c(G ) > (δ(G )− 1)t
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Pursuit-evasion results I.
M. Aigner, M. Fromme (1984):G planar ⇒ c(G ) ≤ 3
A. Quilliot (1985):genus of G : γ(G ) = k⇒ c(G ) ≤ 3 + 2k
P. Frankl (1987):girth of G : g(G ) ≥ 8t − 3⇒ c(G ) > (δ(G )− 1)t
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Pursuit-evasion results II.
M. Aigner, M. Fromme (1984)
Let P be the shortest path between u, v ∈ V (G ). Then one copguarantees that the robber will be immediatelly caught, if hemoves onto P.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Some results:
S. Fitzpatrick, R. Nowakowski (2001):
P = a0a1 . . . an an shortest path in Gf : G → P given by
f (v) ={
ak ; k = d(a0, v) and k ≤ nan ; otherwise
Image or “shadow” of the robber will move along the path.
c(P) = 1⇒ Cop is able to catch a shadow and he can staywith it.
Every vertex on the path is its own shadow ⇒ robber will becaught on P.
H ⊆ G , f : G → H a retraction, H cop-win. A single copcan catch the f -image of the robber in H and stay with it.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Some results:
S. Fitzpatrick, R. Nowakowski (2001):
P = a0a1 . . . an an shortest path in Gf : G → P given by
f (v) ={
ak ; k = d(a0, v) and k ≤ nan ; otherwise
Image or “shadow” of the robber will move along the path.
c(P) = 1⇒ Cop is able to catch a shadow and he can staywith it.
Every vertex on the path is its own shadow ⇒ robber will becaught on P.
H ⊆ G , f : G → H a retraction, H cop-win. A single copcan catch the f -image of the robber in H and stay with it.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Some results:
S. Fitzpatrick, R. Nowakowski (2001):
P = a0a1 . . . an an shortest path in Gf : G → P given by
f (v) ={
ak ; k = d(a0, v) and k ≤ nan ; otherwise
Image or “shadow” of the robber will move along the path.
c(P) = 1⇒ Cop is able to catch a shadow and he can staywith it.
Every vertex on the path is its own shadow ⇒ robber will becaught on P.
H ⊆ G , f : G → H a retraction, H cop-win. A single copcan catch the f -image of the robber in H and stay with it.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Some results:
S. Fitzpatrick, R. Nowakowski (2001):
P = a0a1 . . . an an shortest path in Gf : G → P given by
f (v) ={
ak ; k = d(a0, v) and k ≤ nan ; otherwise
Image or “shadow” of the robber will move along the path.
c(P) = 1⇒ Cop is able to catch a shadow and he can staywith it.
Every vertex on the path is its own shadow ⇒ robber will becaught on P.
H ⊆ G , f : G → H a retraction, H cop-win. A single copcan catch the f -image of the robber in H and stay with it.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Types of guarding
played on graph G = (V , E )for 2 players:
i . . . number of intrudersg . . . number of guards
H subgraph for guarding
first guards step onto G :guarding by protection.
first intruders step onto graph G :guarding by pursuit-evasion.
prevent guards stepping onto the intruder:guard-posts S ⊆ V (G ).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Types of guarding
played on graph G = (V , E )for 2 players:
i . . . number of intrudersg . . . number of guards
H subgraph for guarding
first guards step onto G :guarding by protection.
first intruders step onto graph G :guarding by pursuit-evasion.
prevent guards stepping onto the intruder:guard-posts S ⊆ V (G ).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Types of guarding
played on graph G = (V , E )for 2 players:
i . . . number of intrudersg . . . number of guards
H subgraph for guarding
first guards step onto G :guarding by protection.
first intruders step onto graph G :guarding by pursuit-evasion.
prevent guards stepping onto the intruder:guard-posts S ⊆ V (G ).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Types of guarding
played on graph G = (V , E )for 2 players:
i . . . number of intrudersg . . . number of guards
H subgraph for guarding
first guards step onto G :guarding by protection.
first intruders step onto graph G :guarding by pursuit-evasion.
prevent guards stepping onto the intruder:guard-posts S ⊆ V (G ).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Guarding – formally
played on graph G = (V , E )for 2 players:
i . . . number of intrudersg . . . number of guards
H subgraph for guarding
intruder: (v , ι)
v ∈ V (G )ι : V i+g (G )→ V (G )ι(x , v1, . . . , vi+g−1) ∼ x
guard: (ϕ, γ)
ϕ : V i (G )→ S
γ : V i+g (G )→ V (G )γ(v1, . . . , vi+g−1, y) ∼ y
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Guarding – formally
played on graph G = (V , E )for 2 players:
i . . . number of intrudersg . . . number of guards
H subgraph for guarding
intruder: (v , ι)
v ∈ V (G )ι : V i+g (G )→ V (G )ι(x , v1, . . . , vi+g−1) ∼ x
guard: (ϕ, γ)
ϕ : V i (G )→ S
γ : V i+g (G )→ V (G )γ(v1, . . . , vi+g−1, y) ∼ y
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Guarding – formally
played on graph G = (V , E )for 2 players:
i . . . number of intrudersg . . . number of guards
H subgraph for guarding
intruder: (v , ι)
v ∈ V (G )ι : V i+g (G )→ V (G )ι(x , v1, . . . , vi+g−1) ∼ x
guard: (ϕ, γ)
ϕ : V i (G )→ S
γ : V i+g (G )→ V (G )γ(v1, . . . , vi+g−1, y) ∼ y
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
The length of the game
Prosposition
G . . . graph : n = |V (G )|H . . . subgraph G : n = |V (G )\V (H)|i . . . number of intrudersg . . . number of guards
The intruders will enter H ⇔ intruders will enter H in at most2(ng ni )− 1 moves.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Notation
G a connected graph
H subgraph of G
S ⊆ V (G ) a set of guard posts
W = x0 . . . xl a walk in GY = (ϕ, γ) a guard in G
Position yi of the guard after following the intruder on W :y0(W , Y ) := ϕ(x0)yi (W , Y ) := γ(xi , yi−1)
All forced positions of Y :
ΦY (x) := {yl (W , Y ) |W = x1 . . . xl−1x}W runs over all finite walks in G −H which end in x.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Notation
G a connected graph
H subgraph of G
S ⊆ V (G ) a set of guard posts
W = x0 . . . xl a walk in GY = (ϕ, γ) a guard in G
Position yi of the guard after following the intruder on W :y0(W , Y ) := ϕ(x0)yi (W , Y ) := γ(xi , yi−1)
All forced positions of Y :
ΦY (x) := {yl (W , Y ) |W = x1 . . . xl−1x}W runs over all finite walks in G −H which end in x.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Notation
G a connected graph
H subgraph of G
S ⊆ V (G ) a set of guard posts
W = x0 . . . xl a walk in GY = (ϕ, γ) a guard in G
Position yi of the guard after following the intruder on W :y0(W , Y ) := ϕ(x0)yi (W , Y ) := γ(xi , yi−1)
All forced positions of Y :
ΦY (x) := {yl (W , Y ) |W = x1 . . . xl−1x}W runs over all finite walks in G −H which end in x.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Defending against vertices
Assume: W = x . . . x ′ x ′ ∈ V (H)
A vertex y ∈ V (G ) is said to parry x ∈ V (G ) against thewalk W , if there exists some walk W ′ of length |W | from yto x ′. Notation: y �W x
If y ∈ V (G ) parries x ∈ V (G ) against any walk W from x toany x ′ ∈ V (H), then y parries x . Notation: y � x .
Let Q : V (G )→ 2V (G ) be a function
Q(x) := {y ∈ V (G ) | ∀z ∈ H : d(y , z) ≤ d(x , z)}.Q : V (G )→ 2V (G ) can be computed in O(n2m).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Defending against vertices
Assume: W = x . . . x ′ x ′ ∈ V (H)
A vertex y ∈ V (G ) is said to parry x ∈ V (G ) against thewalk W , if there exists some walk W ′ of length |W | from yto x ′. Notation: y �W x
If y ∈ V (G ) parries x ∈ V (G ) against any walk W from x toany x ′ ∈ V (H), then y parries x . Notation: y � x .
Let Q : V (G )→ 2V (G ) be a function
Q(x) := {y ∈ V (G ) | ∀z ∈ H : d(y , z) ≤ d(x , z)}.Q : V (G )→ 2V (G ) can be computed in O(n2m).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Defending against vertices
Assume: W = x . . . x ′ x ′ ∈ V (H)
A vertex y ∈ V (G ) is said to parry x ∈ V (G ) against thewalk W , if there exists some walk W ′ of length |W | from yto x ′. Notation: y �W x
If y ∈ V (G ) parries x ∈ V (G ) against any walk W from x toany x ′ ∈ V (H), then y parries x . Notation: y � x .
Let Q : V (G )→ 2V (G ) be a function
Q(x) := {y ∈ V (G ) | ∀z ∈ H : d(y , z) ≤ d(x , z)}.Q : V (G )→ 2V (G ) can be computed in O(n2m).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Defending against vertices
Assume: W = x . . . x ′ x ′ ∈ V (H)
A vertex y ∈ V (G ) is said to parry x ∈ V (G ) against thewalk W , if there exists some walk W ′ of length |W | from yto x ′. Notation: y �W x
If y ∈ V (G ) parries x ∈ V (G ) against any walk W from x toany x ′ ∈ V (H), then y parries x . Notation: y � x .
Let Q : V (G )→ 2V (G ) be a function
Q(x) := {y ∈ V (G ) | ∀z ∈ H : d(y , z) ≤ d(x , z)}.Q : V (G )→ 2V (G ) can be computed in O(n2m).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Some properties of �
Reflexive: ∀x : x � x .
Transitive: ∀x , y , z : z � y and y � x implies z � x .
Not antisymmetric: y � x and x � y is possible for x 6= y .
� is a preorder.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Some properties of �
Reflexive: ∀x : x � x .
Transitive: ∀x , y , z : z � y and y � x implies z � x .
Not antisymmetric: y � x and x � y is possible for x 6= y .
� is a preorder.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Some properties of �
Reflexive: ∀x : x � x .
Transitive: ∀x , y , z : z � y and y � x implies z � x .
Not antisymmetric: y � x and x � y is possible for x 6= y .
� is a preorder.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Some properties of �
Reflexive: ∀x : x � x .
Transitive: ∀x , y , z : z � y and y � x implies z � x .
Not antisymmetric: y � x and x � y is possible for x 6= y .
� is a preorder.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
A classification of guarded subgraphs
Theorem
Let G be a graph, H its subgraph and S the set of guard posts.Then H can be guarded in G with a single guard against asingle intruder, if and only if there exists a functionΦ : V (G )→ 2V (G ), such that
Φ(x) ⊆ Q(x) for any x ∈ V (G ),
Φ(x) ∩ S 6= ∅ for any x ∈ V (G ), and
Φ(x) ⊆ Φ+(x ′) for any xx ′ ∈ E (G ).
Proof.
⇒: Y a successful guard: ΦY satisfies these properties. ⇐:y ∈ Φ(x)⇒ y � x . Intruder on x . Initially, the cop moves intoΦ(x) ∩ S , and then stays with Φ(x).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
A classification of guarded subgraphs
Theorem
Let G be a graph, H its subgraph and S the set of guard posts.Then H can be guarded in G with a single guard against asingle intruder, if and only if there exists a functionΦ : V (G )→ 2V (G ), such that
Φ(x) ⊆ Q(x) for any x ∈ V (G ),
Φ(x) ∩ S 6= ∅ for any x ∈ V (G ), and
Φ(x) ⊆ Φ+(x ′) for any xx ′ ∈ E (G ).
Proof.
⇒: Y a successful guard: ΦY satisfies these properties. ⇐:y ∈ Φ(x)⇒ y � x . Intruder on x . Initially, the cop moves intoΦ(x) ∩ S , and then stays with Φ(x).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
A classification of guarded subgraphs
Theorem
Let G be a graph, H its subgraph and S the set of guard posts.Then H can be guarded in G with a single guard against asingle intruder, if and only if there exists a functionΦ : V (G )→ 2V (G ), such that
Φ(x) ⊆ Q(x) for any x ∈ V (G ),
Φ(x) ∩ S 6= ∅ for any x ∈ V (G ), and
Φ(x) ⊆ Φ+(x ′) for any xx ′ ∈ E (G ).
Proof.
⇒: Y a successful guard: ΦY satisfies these properties. ⇐:y ∈ Φ(x)⇒ y � x . Intruder on x . Initially, the cop moves intoΦ(x) ∩ S , and then stays with Φ(x).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
A classification of guarded subgraphs
Theorem
Let G be a graph, H its subgraph and S the set of guard posts.Then H can be guarded in G with a single guard against asingle intruder, if and only if there exists a functionΦ : V (G )→ 2V (G ), such that
Φ(x) ⊆ Q(x) for any x ∈ V (G ),
Φ(x) ∩ S 6= ∅ for any x ∈ V (G ), and
Φ(x) ⊆ Φ+(x ′) for any xx ′ ∈ E (G ).
Proof.
⇒: Y a successful guard: ΦY satisfies these properties. ⇐:y ∈ Φ(x)⇒ y � x . Intruder on x . Initially, the cop moves intoΦ(x) ∩ S , and then stays with Φ(x).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
A classification of guarded subgraphs
Theorem
Let G be a graph, H its subgraph and S the set of guard posts.Then H can be guarded in G with a single guard against asingle intruder, if and only if there exists a functionΦ : V (G )→ 2V (G ), such that
Φ(x) ⊆ Q(x) for any x ∈ V (G ),
Φ(x) ∩ S 6= ∅ for any x ∈ V (G ), and
Φ(x) ⊆ Φ+(x ′) for any xx ′ ∈ E (G ).
Proof.
⇒: Y a successful guard: ΦY satisfies these properties. ⇐:y ∈ Φ(x)⇒ y � x . Intruder on x . Initially, the cop moves intoΦ(x) ∩ S , and then stays with Φ(x).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
A classification of guarded subgraphs
Theorem
Let G be a graph, H its subgraph and S the set of guard posts.Then H can be guarded in G with a single guard against asingle intruder, if and only if there exists a functionΦ : V (G )→ 2V (G ), such that
Φ(x) ⊆ Q(x) for any x ∈ V (G ),
Φ(x) ∩ S 6= ∅ for any x ∈ V (G ), and
Φ(x) ⊆ Φ+(x ′) for any xx ′ ∈ E (G ).
Proof.
⇒: Y a successful guard: ΦY satisfies these properties. ⇐:y ∈ Φ(x)⇒ y � x . Intruder on x . Initially, the cop moves intoΦ(x) ∩ S , and then stays with Φ(x).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
The shadow function
Definition
Inclusion maximal function Φ is called the shadow function of Hin G .
Theorem
Let Φ(x) = {y | y � x}. Then Φ is the inclusion-maximalfunction satisfying the properties from the theorem and can becomputed in O(nm2).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
The shadow function
Definition
Inclusion maximal function Φ is called the shadow function of Hin G .
Theorem
Let Φ(x) = {y | y � x}. Then Φ is the inclusion-maximalfunction satisfying the properties from the theorem and can becomputed in O(nm2).
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
The intruding function
Definition
Let Φ be the shadow function of H in G . DefineΨ : V (G )× V (G )→ 2V (G ) asΨ(x , y) = {z ∈ NG (x) |Φ(z) ∩NG (y) = ∅}. Ψ is called theintruding function of H in G .
Theorem
The following are equivalent for x , y ∈ V (G ):
y � x,
y ∈ Φ(x),
Ψ(x , y) = ∅.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
The intruding function
Definition
Let Φ be the shadow function of H in G . DefineΨ : V (G )× V (G )→ 2V (G ) asΨ(x , y) = {z ∈ NG (x) |Φ(z) ∩NG (y) = ∅}. Ψ is called theintruding function of H in G .
Theorem
The following are equivalent for x , y ∈ V (G ):
y � x,
y ∈ Φ(x),
Ψ(x , y) = ∅.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Correspondences
G , H, x – define the shadow Φ(x) of x .
G , H, x , y – define the intruding function Ψ(x , y).
G , H, x , y – define where the cop should go from y ,Φ(x) ∩N(y).
Retraction – cop’s move depends only on the position of therobber?
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Correspondences
G , H, x – define the shadow Φ(x) of x .
G , H, x , y – define the intruding function Ψ(x , y).
G , H, x , y – define where the cop should go from y ,Φ(x) ∩N(y).
Retraction – cop’s move depends only on the position of therobber?
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Correspondences
G , H, x – define the shadow Φ(x) of x .
G , H, x , y – define the intruding function Ψ(x , y).
G , H, x , y – define where the cop should go from y ,Φ(x) ∩N(y).
Retraction – cop’s move depends only on the position of therobber?
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Correspondences
G , H, x – define the shadow Φ(x) of x .
G , H, x , y – define the intruding function Ψ(x , y).
G , H, x , y – define where the cop should go from y ,Φ(x) ∩N(y).
Retraction – cop’s move depends only on the position of therobber?
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
The shadow and retractions
Reminder
H ⊆ G , f : G → H a homomorphism with f /H = id. Then f is aretraction.
Lema
H ⊆ G , f : G → H a retraction. Then f (x) ∈ Φ(x) for allx ∈ V (G ).
Problem
H ⊆ G , S ⊆ V (G ) a set of guard posts. H can be guarded usingS, if and only if there is a H ′, H ⊆ H ′ ⊆ G [S ], and a retraction ofG onto H ′.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
The shadow and retractions
Reminder
H ⊆ G , f : G → H a homomorphism with f /H = id. Then f is aretraction.
Lema
H ⊆ G , f : G → H a retraction. Then f (x) ∈ Φ(x) for allx ∈ V (G ).
Problem
H ⊆ G , S ⊆ V (G ) a set of guard posts. H can be guarded usingS, if and only if there is a H ′, H ⊆ H ′ ⊆ G [S ], and a retraction ofG onto H ′.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
The shadow and retractions
Reminder
H ⊆ G , f : G → H a homomorphism with f /H = id. Then f is aretraction.
Lema
H ⊆ G , f : G → H a retraction. Then f (x) ∈ Φ(x) for allx ∈ V (G ).
Problem
H ⊆ G , S ⊆ V (G ) a set of guard posts. H can be guarded usingS, if and only if there is a H ′, H ⊆ H ′ ⊆ G [S ], and a retraction ofG onto H ′.
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games
Cops and a rober Guarding a subgraph Single intruder versus single guard
Thank you for your attention!
Drago Bokal, Gordana Radic Guarding a subgraph as a tool in pursuit-evasion games