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

Cops and Robbers on Intersection Graphs ciak , V´ ıt Jel´ ınek, Pavel Klav´ ık, Jan Kratochv´ ıl Tom´ aˇ s Gavenˇ Department of Applied Mathematics, Charles University, Prague Computer Science Institute, Charles University, Prague 31 st 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 genus k string ≤ 2 k + 3 ≤ 30 outer string planar ≤ 30 =3 bounded interval boxicity filament ? =2 chordal circle circular arc function =1 =2 =2 =2 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 v 3 top filaments v 1 v 2 v 4 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 d c = 3 P d r = 3 d c = 3 d c = 5 d r = 4 d r = 5 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 d c = 3 P d r = 4 d c = 3 d c = 5 d r = 3 d r = 4 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 d c = 4 P d r = 4 d c = 2 d c = 4 d r = 3 d r = 4 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 ) ≤ 2 k + 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 cop robber 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 “deputies” cop robber “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 “deputies” cop robber “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. b not a problem (but technical) problem d a c 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. c b a 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. c b a 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. c b a 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