Certain Hypergraphs Pinkaew Siriwong Chulalongkorn University 1983 - - PowerPoint PPT Presentation

Cops and Robbers on Certain Hypergraphs

Pinkaew Siriwong

Chulalongkorn University

Story of Cops and Robbers Game

1983 NOWAKOWSKI AND WINKLER 1984 AIGNER AND FROMME 2011 WILLIAM DAVID BAIRD

• Introduce the game of cops and robbers on graphs
• Characterize cop-win graphs and interested in product of cop-win graphs
• Consider the situation where more cops capture the robber
• For planar graph, three cops suffice to win
• 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

Cops and Robbers

• n Graphs

Start with a (reflexive) finite connected graph Two players: cop and robber

Cop Robber

4

Cops and Robbers

The cop chooses a beginning vertex and then the robber chooses the

• ther vertex to begin

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

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

6

Robber wins

Robber wins if robber can run away (there exists an escaping way for the robber) Example of robber-win graph

7

Cops and Robbers on Hypergraphs

8

Cops and Robbers

• n Hypergraphs

Start with a finite connected hypergraph Two players: cop and robber

Cop Robber

9

Cops and Robbers

The cop chooses a beginning vertex and then the robber chooses the

• ther vertex to begin

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

Cop wins and Robber wins

Example of robber-win hypergraph Example of cop-win hypergraph

11

Cops and Robbers on Products of Hypergraphs

12

Generalization of cops and robbers

1983 (Strong) product of cop-win graphs is cop-win. Consider products of cop-win hypergraphs 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

Cartesian product of cop-win hypergraphs is robber-win



1 2 3 4 5 a b

𝐼1 𝐼2

=

1a 1b 2a 2b 3a 3b 4a 4b 5a 5b The robber always find the vertex to stay far from the cop.

14

Direct Product

Minimal rank preserving direct product of cop-win hypergraphs is robber-win 1 2 3 4 5 a b

𝐼1 𝐼2

=

1a 1b 2a 2b 3a 3b 4a 4b 5a 5b Free neighbor

×1

Free neighbor

15

Direct Product

Maximal rank preserving direct product of cop-win hypergraphs is robber-win 1 2 3 4 5 a b

𝐼1 𝐼2

=

×2

1a 1b 2a 2b 3a 3b 4a 4b 5a 5b Free neighbor

16

Strong Product

1 2 3 4 5 a b

𝐼1 𝐼2

Normal (strong) product of cop-win hypergraphs is cop-win

𝐹 𝐼1 ⊠1 𝐼2 = 𝐹 𝐼1𝐼2 ∪ 𝐹 𝐼1 ×1 𝐼2

⊠𝟐

=

1a 1b 2a 2b 3a 3b 4a 4b 5a 5b

17

Strong Product

1 2 3 4 5 a b

𝐼1

Standard strong product of cop-win hypergraphs is cop-win

𝐹 𝐼1 ⊠2 𝐼2 = 𝐹 𝐼1𝐼2 ∪ 𝐹 𝐼1 ×2 𝐼2

⊠𝟑

1a 1b 2a 2b 3a 3b 4a 4b 5a 5b

=

𝐼2

18

Characterization of Cop-win Hypergraphs

19

Generalization of cops and robbers

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

cop-win graph if and only if ҧ

𝐻 is a cop-win graph.

by successively deletion corners (in any order), 𝐻 can be reduced to 𝐿1

20

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

Characterization of cops and robbers

For some vertex 𝑧 ≠ 𝑦 of a graph 𝐻, 𝑂 𝑦 ⊆ 𝑂 𝑧

𝑦 is a corner of a graph 𝐻.

There exists a vertex 𝑧 of 𝐼 such that 𝑂𝐼 𝑦 ⊆ 𝑂𝐼 𝑧

𝑦 is a corner of a hypergraph 𝐼. 1 is a corner of a graph 𝐻 4 is a corner of a hypergraph 𝐼 𝑂 1 = 1,2,3 𝑂 2 = 1,2,3,5 𝑂𝐼 4 = 4,5,6 𝑂𝐼 6 = 3,4,5,6

21

Characterization of cops and robbers

𝐼 is a robber-win hypergraph if and only if a graph 𝐻 𝐼 of a

hypergraph 𝐼 is a robber-win graph

𝐼 𝐻 𝐼 If 𝐻 𝐼 is a robber-win graph, by applying winning strategy of robber in 𝐻 𝐼 , 𝐼 is a robber-win hypergraph If 𝐼 is a robber-win hypergraph, by applying winning strategy of robber in 𝐼, 𝐻 𝐼 is a robber-win graph

22

𝑊 𝐻 𝐼 = 𝑊 𝐼 and 𝑣𝑤 ∈ 𝐹 𝐻 𝐼

if 𝑣, 𝑤 ⊆ 𝑓 for some 𝑓 ∈ 𝐹 𝐼

Characterization of cops and robbers

Removing of 𝑦 from 𝑊 and removing all edges of 𝐻 containing 𝑦 from 𝐹 Deletion of 𝑦 ∈ 𝑊 from 𝐻 Removing of 𝑦 from 𝑊 𝐼 and from each hyperedge containing 𝑦 Weakly deletion of 𝑦 ∈ 𝑊 𝐼 from 𝐼

23

Characterization of 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 If 𝐼 is a cop-win hypergraph then a graph 𝐻 𝐼 of a hypergraph 𝐼 is a cop-win graph. Thus, 𝐻 𝐼 − 𝑦 is a cop-win graph, so is 𝐻 𝐼 − 𝑦 . Therefore, a weakly deletion 𝐼 − 𝑦 is a cop-win hypergraph.

24

If 𝑦 is a corner in a hypergraph 𝐼, then 𝑦 is a corner in a graph 𝐻 𝐼 .

𝐻 𝐼 − 𝑦 = 𝐻 𝐼 − 𝑦 .

Characterization of 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 If 𝐼 is a robber-win hypergraph then a graph 𝐻 𝐼 of a hypergraph 𝐼 is a robber-win graph. 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

Characterization of cops and robbers

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.

No corner Robber-win hypergraph

26

Characterization of cops and robbers

Reduced to a single vertex Cop-win hypergraph

27

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.

28

Hyperpath and Hpercycle

29

A hypergraph is 𝑢-joined if each intersection of hyperedges contains exactly 𝑢 vertices A hyperpath is a sequence of hyperedges 𝐹1, 𝐹2, 𝐹3, … , 𝐹𝑙, such that 𝐹𝑗 and 𝐹𝑗+1 are 𝑢-joined for 𝑢 > 0 and for 1 ≤ 𝑗 ≤ 𝑙 − 1 and 𝐹𝑗 ∩ 𝐹

𝑘 = ∅ when 𝑘 ≠ 𝑗 + 1 𝑛𝑝𝑒 𝑙

For an integer 𝑙 > 2, a 𝑙-hypercycle is a collection of 𝑙 hyperedges 𝐹1, 𝐹2, 𝐹3, … , 𝐹𝑙, with two hyperedges 𝐹𝑗 and 𝐹

𝑘

are incident if 𝑘 = 𝑗 + 1 𝑛𝑝𝑒 𝑙

➢ Vertex-set: 𝑊

1 × 𝑊 2

• 1. 𝑦1, 𝑦2, 𝑦3, … , 𝑦𝑠 ∈ 𝐹1 and 𝑧1 = 𝑧2 = 𝑧3 = ⋯ = 𝑧𝑠 ∈ 𝑊

2

• 2. 𝑧1, 𝑧2, 𝑧3, … , 𝑧𝑠 ∈ 𝐹2 and 𝑦1 = 𝑦2 = 𝑦3 = ⋯ = 𝑦𝑠 ∈ 𝑊

1

➢ Edge-set: 𝑦1, 𝑧1 , 𝑦2, 𝑧2 , 𝑦3, 𝑧3 , … , 𝑦𝑠, 𝑧𝑠 is an edge if

30

The products of

𝐼1 𝑊

1,𝐹1 and

𝐼2 𝑊

2,𝐹2

The Cartesian Product 𝑰𝟐𝑰𝟑

➢ Vertex-set: 𝑊

1 × 𝑊 2

➢ Edge-set: 𝑦1, 𝑧1 , 𝑦2, 𝑧2 , 𝑦3, 𝑧3 , … , 𝑦𝑠, 𝑧𝑠 is an edge if

31

The products of

𝐼1 𝑊

1,𝐹1 and

𝐼2 𝑊

2,𝐹2

The Minimal Rank Preserving Direct Product 𝑰𝟐 ×𝟐 𝑰𝟑

• 1. 𝑦1, 𝑦2, 𝑦3, … , 𝑦𝑠 ∈ 𝐹1 and 𝑧1, 𝑧2, 𝑧3, … , 𝑧𝑠 ⊆ 𝑓2 for some 𝑓2 ∈ 𝐹2
• 2. 𝑦1, 𝑦2, 𝑦3, … , 𝑦𝑠 ⊆ 𝑓1 for some 𝑓1 ∈ 𝐹1 and 𝑧1, 𝑧2, 𝑧3, … , 𝑧𝑠 ∈ 𝐹2

➢ Vertex-set: 𝑊

1 × 𝑊 2

➢ Edge-set: 𝑦1, 𝑧1 , 𝑦2, 𝑧2 , 𝑦3, 𝑧3 , … , 𝑦𝑠, 𝑧𝑠 is an edge if

32

The products of

𝐼1 𝑊

1,𝐹1 and

𝐼2 𝑊

2,𝐹2

The Maximal Rank Preserving Direct Product 𝑰𝟐 ×𝟑 𝑰𝟑

• 1. 𝑦1, 𝑦2, 𝑦3, … , 𝑦𝑠 ∈ 𝐹1 and there is an edge 𝑓2 ∈ 𝐹2 such that

𝑧1, 𝑧2, 𝑧3, … , 𝑧𝑠 is a multiset of elements of 𝑓2, and 𝑓2 ⊆ 𝑧1, 𝑧2, 𝑧3, … , 𝑧𝑠

• 2. 𝑧1, 𝑧2, 𝑧3, … , 𝑧𝑠 ∈ 𝐹2 and there is an edge 𝑓1 ∈ 𝐹1 such that

𝑦1, 𝑦2, 𝑦3, … , 𝑦𝑠 is a multiset of elements of 𝑓1, and 𝑓1 ⊆ 𝑦1, 𝑦2, 𝑦3, … , 𝑦𝑠