Cops and Robbers on Intersection Graphs ciak , V t Jel nek, Pavel - - PowerPoint PPT Presentation

Cops and Robbers on Intersection Graphs

Tom´ aˇ s Gavenˇ ciak, V´ ıt Jel´ ınek, Pavel Klav´ ık, Jan Kratochv´ ıl

Department of Applied Mathematics, Charles University, Prague Computer Science Institute, Charles University, Prague

31st Mar 2014 Grasta 2014

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Cops and robber game

One player controls k cops, the other one one robber; moving on vertices of graph G.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Cops and robber game

One player controls k cops, the other one one robber; moving on vertices of graph G.

Game start

◮ First player places k cops. ◮ The second player places the robber.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Cops and robber game

One player controls k cops, the other one one robber; moving on vertices of graph G.

Game start

◮ First player places k cops. ◮ The second player places the robber.

One turn

◮ Every cop moves to distance at most 1. ◮ The robber moves to distance at most 1.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Cops and robber game

One player controls k cops, the other one one robber; moving on vertices of graph G.

Game start

◮ First player places k cops. ◮ The second player places the robber.

One turn

◮ Every cop moves to distance at most 1. ◮ The robber moves to distance at most 1.

Victory

◮ The cops win if a cop shares a vertex with the robber. ◮ The robber wins by escaping indefinitely.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Cop-number bounds on some classes

interval filament =2

• uter string

≤ 30 string ≤ 30 genus k ≤ 2k + 3 planar =3 function =2 circular arc =2 circle =2 bounded boxicity ? chordal =1 interval =1

We assume connected graphs.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on IFA graphs

IFA graphs [Gavril 2000]

Intersection graphs of interval filaments.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on IFA graphs

IFA graphs [Gavril 2000]

Intersection graphs of interval filaments.

Theorem

On IFA graphs we have cn(G) ≤ 2.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on IFA graphs

cop

Theorem

On IFA graphs we have cn(G) ≤ 2.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on IFA graphs

cop

Theorem

On IFA graphs we have cn(G) ≤ 2.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on IFA graphs

cop

Theorem

On IFA graphs we have cn(G) ≤ 2.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on IFA graphs

cop v1 v2 v3 v4 top filaments

Theorem

On IFA graphs we have cn(G) ≤ 2.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on IFA graphs

cop cop

Theorem

On IFA graphs we have cn(G) ≤ 2.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on IFA graphs

cop cop

Theorem

On IFA graphs we have cn(G) ≤ 2.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on IFA graphs

cop cop

Theorem

On IFA graphs we have cn(G) ≤ 2.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on planar graphs

Theorem (Aigner, Fromme 1984)

In any graph, one cop can prevent the robber from entering a given shortest path P.

robber cop dc = 3 dr = 3 dc = 5 dr = 5 dc = 3 dr = 4 P

Cop: Keep every vertex of P at least as close to you as to the robber.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on planar graphs

Theorem (Aigner, Fromme 1984)

In any graph, one cop can prevent the robber from entering a given shortest path P.

robber cop dc = 3 dr = 4 dc = 5 dr = 4 dc = 3 dr = 3 P

Cop: Keep every vertex of P at least as close to you as to the robber.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on planar graphs

Theorem (Aigner, Fromme 1984)

In any graph, one cop can prevent the robber from entering a given shortest path P.

robber cop dc = 4 dr = 4 dc = 4 dr = 4 dc = 2 dr = 3 P

Cop: Keep every vertex of P at least as close to you as to the robber.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on planar graphs

Theorem (Aigner, Fromme 1984)

In any graph, one cop can prevent the robber from entering a given shortest path P.

Theorem (Aigner, Fromme 1984)

For planar graphs, cn(G) ≤ 3. This bound is sharp.

Theorem (Aigner, Fromme 1984)

For genus-k graphs, cn(G) ≤ 2k + 3.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

String graphs

String graph = intesrection graph of simple stings in the plane.

strings in plane intersection graph

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Guarding paths: string graphs

In string graphs, guarding P is not sufficient to prevent robber from crossing P.

P robber cop

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Guarding paths: string graphs

In string graphs, guarding P is not sufficient to prevent robber from crossing P.

P robber cop “deputies” “deputies”

Solution: Also guard two previous and following vertices of P.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Guarding paths: string graphs

In string graphs, guarding P is not sufficient to prevent robber from crossing P.

P robber cop “deputies” “deputies”

Solution: Also guard two previous and following vertices of P.

Theorem

In any graph, five cops can prevent the robber from entering neighbourhood of a given shortest path P.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on string graphs

Theorem

For any string graph G we have cn(G) ≤ 30. This implies bounded cn for many intersecton graphs of connected planar objects.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on string graphs

Theorem

For any string graph G we have cn(G) ≤ 30.

R

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on string graphs

Theorem

For any string graph G we have cn(G) ≤ 30.

R

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on string graphs

Theorem

For any string graph G we have cn(G) ≤ 30.

R R R R

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on string graphs

Theorem

For any string graph G we have cn(G) ≤ 30.

R R R R

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

C&R on string graphs

Theorem

For any string graph G we have cn(G) ≤ 30. Alas, string graphs may have complicated shortest paths.

a b c d problem not a problem (but technical)

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Trimming string graphs

Cops will make sure the robber never leaves the blue area.

a b c

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Trimming string graphs

Removing parts of strings may split vertices and delete edges.

a b c

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Trimming string graphs

Lemma

A strategy in trimmed graph limiting the robber to the inside also gives a strategy for the untrimmed graph.

a b c

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Conclusion

Observation

When a graph class has cn(G) bounded, computing cn(G) is polynomial.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Conclusion

Observation

When a graph class has cn(G) bounded, computing cn(G) is polynomial.

Open problems

◮ Bound cn(G) of other graph classes. ◮ Sharp bounds for cn(G) of genus-k graphs. ◮ Better bounds for cn(G) of string and outer-string graphs. ◮ Give better algorithms for cn(G) than observed above.

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs

Thank you!

Gavenˇ ciak, Jel´ ınek, Klav´ ık, Kratochv´ ıl Cops and Robbers on Intersection Graphs