Cops and Robbers on Certain Hypergraphs Pinkaew Siriwong Chulalongkorn University
1983 NOWAKOWSKI AND WINKLER • Introduce the game of cops and robbers on graphs • Characterize cop-win graphs and interested in product of cop-win graphs 1984 AIGNER AND FROMME Story of Cops and Robbers Game • Consider the situation where more cops capture the robber • For planar graph, three cops suffice to win 2011 WILLIAM DAVID BAIRD • Introduce the game of cops and robbers on hypergraphs • Investigate that hyperpath is cop-win, but hypercycle is robber-win 2
Cops and Robbers on Graphs 3
Start with a (reflexive) finite connected graph Cops and Robbers on Graphs Two players: cop and robber Cop Robber 4
The cop chooses a beginning vertex and then the robber chooses the other vertex to begin Cops and Robbers In each round, the cop and the robber take alternatively moving from their present vertex to other vertices along edges or staying put 5
Cop wins if cop can catch robber by occupying the same vertex as the robber after finite number of moves Example of cop-win graph Cop wins 6
Robber wins if robber can run away (there exists an escaping way for the robber) Example of robber-win graph Robber wins 7
Cops and Robbers on Hypergraphs 8
Start with a finite connected hypergraph Cops and Robbers on Hypergraphs Two players: cop and robber Cop Robber 9
The cop chooses a beginning vertex and then the robber chooses the other vertex to begin Cops and Robbers In each round, they take alternatively moving from their present vertex 𝑦 to any vertex 𝑧 belonging to the same hyperedge as vertex 𝑦 or staying put. 10
Example of cop-win hypergraph Cop wins and Robber wins Example of robber-win hypergraph 11
Cops and Robbers on Products of Hypergraphs 12
1983 (Strong) product of cop-win graphs is cop-win. Consider products of cop-win hypergraphs Generalization of cops and robbers Cartesian product of cop-win hypergraphs is robber-win Direct product of cop-win hypergraphs is robber-win Strong product of cop-win hypergraphs is cop-win 13
Cartesian product of cop-win hypergraphs is robber-win a 4 1 3 2 5 b 𝐼 1 Cartesian Product 𝐼 2 1a 2a 3a 4a 5a = 1b 2b 3b 4b 5b The robber always find the vertex to stay far from the cop. 14
Minimal rank preserving direct product of cop-win hypergraphs is robber-win a × 1 4 1 3 2 5 b 𝐼 1 Direct Product Free neighbor 𝐼 2 1a 2a 3a 4a 5a = 1b 2b 3b 4b 5b Free neighbor 15
Maximal rank preserving direct product of cop-win hypergraphs is robber-win a × 2 4 1 3 2 5 b 𝐼 1 Direct Product Free neighbor 𝐼 2 1a 2a 3a 4a 5a = 1b 2b 3b 4b 5b 16
𝐹 𝐼 1 ⊠ 1 𝐼 2 = 𝐹 𝐼 1 𝐼 2 ∪ 𝐹 𝐼 1 × 1 𝐼 2 Normal (strong) product of cop-win hypergraphs is cop-win a 4 ⊠ 𝟐 1 3 2 5 b 𝐼 1 Strong Product 𝐼 2 1a 2a 3a 4a 5a = 1b 2b 3b 4b 5b 17
𝐹 𝐼 1 ⊠ 2 𝐼 2 = 𝐹 𝐼 1 𝐼 2 ∪ 𝐹 𝐼 1 × 2 𝐼 2 Standard strong product of cop-win hypergraphs is cop-win a 4 ⊠ 𝟑 1 3 2 5 b 𝐼 1 Strong Product 𝐼 2 1a 2a 3a 4a 5a = 1b 2b 3b 4b 5b 18
Characterization of Cop-win Hypergraphs 19
by successively deletion corners (in any order), 𝐻 can be reduced to 𝐿 1 1983 A finite cop-win graph if and only if it is dismantlable . 1984 Let 𝑦 be a corner of 𝐻 and ҧ 𝐻 = 𝐻 − 𝑦 . 𝐻 is a Generalization of cop-win graph if and only if ҧ 𝐻 is a cop-win graph. cops and robbers Let 𝑦 be a corner of a hypergraph 𝐼 . 𝐼 is a cop-win hypergraph if and only if a weakly deletion 𝐼 − 𝑦 is a cop-win hypergraph 20
𝑦 is a corner of a hypergraph 𝐼 . 𝑦 is a corner of a graph 𝐻 . There exists a vertex 𝑧 of 𝐼 such For some vertex 𝑧 ≠ 𝑦 of a graph 𝐻 , 𝑂 𝑦 ⊆ 𝑂 𝑧 that 𝑂 𝐼 𝑦 ⊆ 𝑂 𝐼 𝑧 Characterization of 𝑂 2 = 1,2,3,5 𝑂 1 = 1,2,3 cops and robbers 𝑂 𝐼 4 = 4,5,6 𝑂 𝐼 6 = 3,4,5,6 1 is a corner of a graph 𝐻 4 is a corner of a hypergraph 𝐼 21
= 𝑊 𝐼 and 𝑣𝑤 ∈ 𝐹 𝐻 𝐼 𝑊 𝐻 𝐼 if 𝑣, 𝑤 ⊆ 𝑓 for some 𝑓 ∈ 𝐹 𝐼 𝐼 is a robber-win hypergraph if and only if a graph 𝐻 𝐼 of a hypergraph 𝐼 is a robber-win graph Characterization of cops and robbers 𝐼 𝐻 𝐼 If 𝐼 is a robber-win hypergraph, by applying winning strategy of robber in 𝐼 , 𝐻 𝐼 is a robber-win graph If 𝐻 𝐼 is a robber-win graph, by applying winning strategy of robber in 𝐻 𝐼 , 𝐼 is a robber-win hypergraph 22
Deletion of 𝑦 ∈ 𝑊 from 𝐻 Weakly deletion of 𝑦 ∈ 𝑊 𝐼 from 𝐼 Removing of 𝑦 from 𝑊 and removing Removing of 𝑦 from 𝑊 𝐼 and all edges of 𝐻 containing 𝑦 from 𝐹 from each hyperedge containing 𝑦 Characterization of cops and robbers 23
Let 𝑦 be a corner of a hypergraph 𝐼 . 𝐼 is a cop-win hypergraph if and only if a weakly deletion 𝐼 − 𝑦 is a cop-win hypergraph If 𝐼 is a cop-win hypergraph then a graph 𝐻 𝐼 of a hypergraph 𝐼 is a cop-win graph. Characterization of cops and robbers Thus, 𝐻 𝐼 − 𝑦 is a cop-win graph, so is 𝐻 𝐼 − 𝑦 . 𝐻 𝐼 − 𝑦 = 𝐻 𝐼 − 𝑦 . If 𝑦 is a corner in a hypergraph 𝐼 , then 𝑦 is a corner in a graph 𝐻 𝐼 . Therefore, a weakly deletion 𝐼 − 𝑦 is a cop-win hypergraph. 24
Let 𝑦 be a corner of a hypergraph 𝐼 . 𝐼 is a cop-win hypergraph if and only if a weakly deletion 𝐼 − 𝑦 is a cop-win hypergraph If 𝐼 is a robber-win hypergraph then a graph 𝐻 𝐼 of a hypergraph 𝐼 is a robber-win graph. Characterization of cops and robbers Thus, 𝐻 𝐼 − 𝑦 is a robber-win graph, so is 𝐻 𝐼 − 𝑦 . Therefore, a weakly deletion 𝐼 − 𝑦 is a robber-win hypergraph. 25
A hypergraph 𝐼 is a cop-win hypergraph if and only if by successively weakly deletion corners (in any order), 𝐼 can be reduced to a single vertex. Characterization of cops and robbers No corner Robber-win hypergraph 26
A hypergraph 𝐼 is a cop-win hypergraph if and only if by successively weakly deletion corners (in any order), 𝐼 can be reduced to a single vertex. Characterization of cops and robbers Reduced to a single vertex Cop-win hypergraph 27
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A hypergraph is 𝑢 -joined if each intersection of hyperedges contains exactly 𝑢 vertices A hyperpath is a sequence of hyperedges 𝐹 1 , 𝐹 2 , 𝐹 3 , … , 𝐹 𝑙 , such that 𝐹 𝑗 and 𝐹 𝑗+1 are 𝑢 -joined for 𝑢 > 0 and for 1 ≤ 𝑗 ≤ 𝑙 − 1 Hyperpath and and 𝐹 𝑗 ∩ 𝐹 𝑘 = ∅ when 𝑘 ≠ 𝑗 + 1 𝑛𝑝𝑒 𝑙 Hpercycle For an integer 𝑙 > 2 , a 𝑙 -hypercycle is a collection of 𝑙 hyperedges 𝐹 1 , 𝐹 2 , 𝐹 3 , … , 𝐹 𝑙 , with two hyperedges 𝐹 𝑗 and 𝐹 𝑘 are incident if 𝑘 = 𝑗 + 1 𝑛𝑝𝑒 𝑙 29
The Cartesian Product 𝑰 𝟐 𝑰 𝟑 ➢ Vertex-set: 𝑊 1 × 𝑊 2 The products of 1 ,𝐹 1 and 𝐼 1 𝑊 ➢ Edge-set: 𝑦 1 , 𝑧 1 , 𝑦 2 , 𝑧 2 , 𝑦 3 , 𝑧 3 , … , 𝑦 𝑠 , 𝑧 𝑠 is an edge 𝐼 2 𝑊 2 ,𝐹 2 if 1. 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑠 ∈ 𝐹 1 and 𝑧 1 = 𝑧 2 = 𝑧 3 = ⋯ = 𝑧 𝑠 ∈ 𝑊 2 2. 𝑧 1 , 𝑧 2 , 𝑧 3 , … , 𝑧 𝑠 ∈ 𝐹 2 and 𝑦 1 = 𝑦 2 = 𝑦 3 = ⋯ = 𝑦 𝑠 ∈ 𝑊 1 30
The Minimal Rank Preserving Direct Product 𝑰 𝟐 × 𝟐 𝑰 𝟑 ➢ Vertex-set: 𝑊 1 × 𝑊 2 The products of 1 ,𝐹 1 and 𝐼 1 𝑊 ➢ Edge-set: 𝑦 1 , 𝑧 1 , 𝑦 2 , 𝑧 2 , 𝑦 3 , 𝑧 3 , … , 𝑦 𝑠 , 𝑧 𝑠 is an edge 𝐼 2 𝑊 2 ,𝐹 2 if 1. 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑠 ∈ 𝐹 1 and 𝑧 1 , 𝑧 2 , 𝑧 3 , … , 𝑧 𝑠 ⊆ 𝑓 2 for some 𝑓 2 ∈ 𝐹 2 2. 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑠 ⊆ 𝑓 1 for some 𝑓 1 ∈ 𝐹 1 and 𝑧 1 , 𝑧 2 , 𝑧 3 , … , 𝑧 𝑠 ∈ 𝐹 2 31
The Maximal Rank Preserving Direct Product 𝑰 𝟐 × 𝟑 𝑰 𝟑 ➢ Vertex-set: 𝑊 1 × 𝑊 2 The products of 1 ,𝐹 1 and 𝐼 1 𝑊 ➢ Edge-set: 𝑦 1 , 𝑧 1 , 𝑦 2 , 𝑧 2 , 𝑦 3 , 𝑧 3 , … , 𝑦 𝑠 , 𝑧 𝑠 is an edge 𝐼 2 𝑊 2 ,𝐹 2 1. 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑠 ∈ 𝐹 1 and there is an edge 𝑓 2 ∈ 𝐹 2 such that if 𝑧 1 , 𝑧 2 , 𝑧 3 , … , 𝑧 𝑠 is a multiset of elements of 𝑓 2 , and 𝑓 2 ⊆ 𝑧 1 , 𝑧 2 , 𝑧 3 , … , 𝑧 𝑠 2. 𝑧 1 , 𝑧 2 , 𝑧 3 , … , 𝑧 𝑠 ∈ 𝐹 2 and there is an edge 𝑓 1 ∈ 𝐹 1 such that 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑠 is a multiset of elements of 𝑓 1 , and 𝑓 1 ⊆ 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑠 32
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