Guarding a subgraph as a tool in pursuit-evasion games · Guarding a subgraph as a tool in...

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


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



Basic rules of the game

played on a connected graph G = (V , E )2 players: cop, robber

perfect information

cops move, robber moves





perfect information






perfect information






perfect information






perfect information






perfect information






perfect information






perfect information




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.




cop-number: c(G )



tree is cop-win







cop-number: c(G )



tree is cop-win







cop-number: c(G )



tree is cop-win






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















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.



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.



Some results:



f (v) ={








Some results:



f (v) ={








Some results:



f (v) ={








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 ).



Types of guarding









Types of guarding









Types of guarding









Guarding – formally




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







intruder: (v , ι)


guard: (ϕ, γ)

ϕ : V i (G )→ S

γ : V i+g (G )→ V (G )γ(v1, . . . , vi+g−1, y) ∼ y







intruder: (v , ι)


guard: (ϕ, γ)

ϕ : V i (G )→ S

γ : V i+g (G )→ V (G )γ(v1, . . . , vi+g−1, y) ∼ y



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.



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.



Notation

G a connected graph

H subgraph of G








Notation

G a connected graph

H subgraph of G








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).




Assume: W = x . . . x ′ x ′ ∈ V (H)








Assume: W = x . . . x ′ x ′ ∈ V (H)








Assume: W = x . . . x ′ x ′ ∈ V (H)







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.







� is a preorder.







� is a preorder.







� is a preorder.



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).




Theorem




Φ(x) ⊆ Φ+(x ′) for any xx ′ ∈ E (G ).

Proof.





Theorem




Φ(x) ⊆ Φ+(x ′) for any xx ′ ∈ E (G ).

Proof.





Theorem




Φ(x) ⊆ Φ+(x ′) for any xx ′ ∈ E (G ).

Proof.





Theorem




Φ(x) ⊆ Φ+(x ′) for any xx ′ ∈ E (G ).

Proof.





Theorem




Φ(x) ⊆ Φ+(x ′) for any xx ′ ∈ E (G ).

Proof.




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).



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).



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) = ∅.



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) = ∅.



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?



Correspondences







Correspondences







Correspondences







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 ′.




Reminder


Lema


Problem





Reminder


Lema


Problem




Thank you for your attention!


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