Connected treewidth and connected cops-and-robber game - PowerPoint PPT Presentation

Connected treewidth and connected cops-and-robber game – Obstructions and algorithms Christophe PAUL (CNRS – Univ. Montpellier, LIRMM, France) Joint work with I. Adler (University of Leeds, UK) G. Mescoff (ENS Rennes, France) D. Thilikos (CNRS – Univ. Montpellier, LIRMM, France) CAALM Workshop, Chennai, January 25, 2019

A node search strategy A search strategy is defined by a sequence of moves, each of these ◮ either add a searcher d e a b c f g

A node search strategy A search strategy is defined by a sequence of moves, each of these ◮ either add a searcher d e �{ a } , . . . � a b c f g

A node search strategy A search strategy is defined by a sequence of moves, each of these ◮ either add a searcher d e �{ a } , { a , b } , . . . � a b c f g

A node search strategy A search strategy is defined by a sequence of moves, each of these ◮ either add a searcher ◮ or remove a searcher d e �{ a } , { a , b } , { b } , . . . � a b c f g More formally, we define S = � S 1 , . . . S r � such that ◮ for all i ∈ [ r ], S i ⊆ V ( G ); (set of occupied positions) ◮ | S 1 | = 1; ◮ for all i ∈ [ r − 1], | S i △ S i − 1 | = 1.

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ ??? Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ Lazy robber Agile robber We define the set of free locations in the case of a lazy robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i ∩ ( S i \ S i − 1 ) }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ agile : he can move (if possible) at any time ??? Lazy robber Agile robber We define the set of free locations in the case of a agile robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ agile : he can move (if possible) at any time Lazy robber Agile robber We define the set of free locations in the case of a agile robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ agile : he can move (if possible) at any time Lazy robber Agile robber We define the set of free locations in the case of a agile robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ agile : he can move (if possible) at any time Lazy robber Agile robber We define the set of free locations in the case of a agile robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ agile : he can move (if possible) at any time Lazy robber Agile robber We define the set of free locations in the case of a agile robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i }

Node search against. . . . . . an invisible robber, that can be ◮ lazy : he escapes (if possible) if a searcher is landing at his position ◮ agile : he can move (if possible) at any time Lazy robber Agile robber We define the set of free locations in the case of a agile robber : ◮ F 1 = V ( G ) \ S 1 ◮ for all i � 2, F i = ( F i − 1 \ S i ) ∪ { v ∈ cc G − S i ( u ) | u ∈ F i }

Properties and cost of a node search strategy A node search strategy S = � S 1 , . . . S r � is ◮ complete if F r = ∅ ; ◮ monotone if for every i ∈ [ r − 1], F i +1 ⊂ F i . (there is no recontamination of a vertex)

Properties and cost of a node search strategy A node search strategy S = � S 1 , . . . S r � is ◮ complete if F r = ∅ ; ◮ monotone if for every i ∈ [ r − 1], F i +1 ⊂ F i . (there is no recontamination of a vertex) We define ans ( G ) = min { cost( S ) | S is a complete strategy against an agile robber } mans ( G ) = min { cost( S ) | S is a complete monotone . . . agile robber } lns ( G ) = min { cost( S ) | S is a complete strategy against a lazy robber } mlns ( G ) = min { cost( S ) | S is a complete monotone . . . lazy robber }