Domination Game

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Domination Game

Sandi Klavzar

Faculty of Mathematics and Physics, University of Ljubljana, Slovenia

Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia

Institute of Mathematics, Physics and Mechanics, Ljubljana

Midsummer Combinatorial Workshop XIX

July 29 - August 2, 2013, Prague

Domination Game
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The game

Domination Game
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Domination game on G

For a graph G= (V, E), thedomination numberofG is theminimum number, denoted (G), of vertices in a subset A ofV such that V =N[A] =xAN[x].

Domination Game
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Two players, Dominator (D) and Staller (S), take turns choosing a

vertex in G.

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vertex in G.IfCdenotes the set of vertices chosen at some point in agame and D orSchooses vertex w, then N[w] N[C]=.

Domination Game
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vertex in G.IfCdenotes the set of vertices chosen at some point in agame and D orSchooses vertex w, then N[w] N[C]=.

D uses a strategy to end the game in as few moves aspossible;Suses a strategy that will require the most movesbefore the game ends.

Domination Game
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Game domination numberThegame domination number ofG is the number of moves,g(G), when D moves first and both players use an optimalstrategy. (Game 1)

Domination Game
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d1, s1, d2, s2, . . . denotes the sequence of moves.

Domination Game
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Thestaller-start game domination numberofG is the numberof moves, g(G), when Smoves first and both players use anoptimal strategy. (Game 2)

Domination Game
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Thestaller-start game domination numberofG is the numberof moves, g(G), when Smoves first and both players use anoptimal strategy. (Game 2)

s1, d

1, s

2, d

2, . . . denotes the sequence of moves.

Domination Game

Th P
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The game on P5

Th P
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The game on P5

d1

Th P
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The game on P5

d1 s1

Th g P
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The game on P5

d1 s1

d2

The game on P
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The game on P5

d1

s1 d2

The game on P
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The game on P5

d1

s1 d2

d1

The game on P
http://goforward/http://find/http://goback/


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The game on P5

d1

s1 d2

d1 s1

The game on P5
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The game on P5

d1

s1 d2

d1 s1 d2

The game on P5
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The game on P5

d1 s1 d2

d1 s1 d2

The game on P5
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The game on P5

d1 s1 d2

d1 s1 d2

d1

The game on P5
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The game on P5

d1 s1 d2

d1 s1 d2

d1s1

The game on P5
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The game on P5

d1 s1 d2

d1 s1 d2

d1s1 d2

The game on P5
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The game on P5

d1 s1 d2

d1 s1 d2

d1s1 d2

g(P5) = 3

Domination Game

The game on P5 contd
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g 5

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g 5

s1

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g 5

s1d1

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g 5

s1d1

g(P5) = 3 =

g(P5)

Domination Game

The game on a tree
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g

The game on a tree
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d1

The game on a tree
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d1

s1

The game on a tree
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d1

s1

The game on a tree
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d1

s1

s1

The game on a tree
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d1

s1

s1

d1

The game on a tree
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d1

s1

s1

d1

s2

The game on a tree
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d1

s1

s1

d1

s2

g(T) = 2, g(T) = 3

Domination Game

The game on C6
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The game on C6
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d1

The game on C6
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d1

s1

The game on C6
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d1

s1

d2
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The game on C6


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d1

s1

d2

s1

The game on C6
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d1

s1

d2

s1d1

The game on C6
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d1

s1

d2

s1d1

g(C6) = 3, g(C6) = 2

Domination Game
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g versus


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SupposeD and Splay Game 1 on G.

Domination Game

g versus
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Let Cdenote the total set of vertices chosen by D and S.Then C is a dominating set ofG.

Domination Game

g versus
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IfD employs a strategy of selecting vertices from a minimumdominating set A ofG, then Game 1 will have ended when D

has exhausted the vertices from A.

Domination Game

g versus
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IfD employs a strategy of selecting vertices from a minimumdominating set A ofG, then Game 1 will have ended when D

has exhausted the vertices from A.

Theorem (Bresar, K., Rall, 2010)

If G is any graph, then (G) g(G) 2(G) 1. Moreover, forany integer k1 and any0 rk 1, there exists a graph Gwith (G) =k andg(G) =k+r .

Domination Game

g versus g
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Theorem (Bresar, K., Rall, 2010; Kinnersley, West, Zamani, 2013?)

For any graph G, |g(G) g(G)| 1.

Domination Game

g versus g
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Apartially dominated graphis a graph in which some verticeshave already been dominated in some turns of the gamealready played.

Domination Game

g versus g
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IfX is a partially dominated graph, then g(X) (g(X)) is thenumber of turns remaining ifD (S) has the move.

Domination Game

g versus g
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IfX is a partially dominated graph, then g(X) (g(X)) is thenumber of turns remaining ifD (S) has the move.

Lemma (Kinnersley, West, Zamani, 2013?)

(Continuation Principle)Let G be a graph and A, BV(G). LetGA and GBbe partially dominated graphs in which the sets A andB have already been dominated, respectively. If BA, theng(GA) g(GB) and

g(GA)

g(GB).

Domination Game

Proof of Continuation Principle
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D will play two games: Game A on GA (real game) and GameB onGB(imagined game).

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D will keepthe rule: the set of vertices dominated in Game Ais a superset of vertices dominated in Game B.

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Suppose Game B is not yet finished. If there are noundominated vertices in Game A, then Game A has finishedbefore Game B and we are done.

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Suppose Game B is not yet finished. If there are noundominated vertices in Game A, then Game A has finishedbefore Game B and we are done.

It is Ds move: he selects an optimal move in game B. If it islegal in Game A, he plays it there as well, otherwise he plays

any undominated vertex.

Domination Game

Proof of Continuation Principle contd
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It is Ss move: she plays in Game A. Bythe rule, this move islegal in Game B and D can replicate it in Game B.

Domination Game

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It is Ss move: she plays in Game A. Bythe rule, this move islegal in Game B and D can replicate it in Game B.

Bythe rule, Game A finishes no later than Game B.

Domination Game

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It is Ss move: she plays in Game A. Bythe rule, this move is

legal in Game B and D can replicate it in Game B.


D played optimally on Game B. Hence:IfD played first in Game B, the number of moves taken on

Game B was at most g(GB) (indeed, ifSdid not playoptimally, it might be strictly less);IfSplayed first in Game B, the number of moves taken onGame B was at most g(GB).

Domination Game

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It is Ss move: she plays in Game A. Bythe rule, this move is

legal in Game B and D can replicate it in Game B.


D played optimally on Game B. Hence:IfD played first in Game B, the number of moves taken on

Game B was at most g(GB) (indeed, ifSdid not playoptimally, it might be strictly less);IfSplayed first in Game B, the number of moves taken onGame B was at most g(GB).

Hence

IfD played first in Game B, then g(GA) g(GB);IfSplayed first in Game B, then g(GA)

g(GB).

Domination Game

Proof of the theorem
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Consider Game 1 and let vbe the first move ofD.

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Let G be the resulting partially dominated graph.

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g(G) g(G) + 1.

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g(G) g(G) + 1.

By Continuation Principle, g(G) g(G).

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g(G) g(G) + 1.

By Continuation Principle, g(G) g(G).

Hence g(G) g(G) + 1 g(G) + 1.

By a parallel argument, g(G) g(G) + 1.

Domination Game

Realizable pairs
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A pair (r, s) of integers is realizableif there exists a graph G such

that g(G) =r and g(G) =s.

Domination Game

Realizable pairs
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that g(G) =r and g(G) =s. By the theorem, only possiblerealizable pairs are: (r, r), (r, r+ 1), (r, r 1).

Domination Game

Realizable pairs
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Theorem (Kosmrlj, 2014)

Pairs(r, r), r2, (r, r+ 1), r1, and(2k, 2k 1), k2, arerealizable by 2-connected graphs. Pairs(2k+ 1, 2k), k2arerealizable by connected graphs.

Domination Game

Realizable pairs
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Theorem (Kosmrlj, 2014)

Pairs(r, r), r2, (r, r+ 1), r1, and(2k, 2k 1), k2, arerealizable by 2-connected graphs. Pairs(2k+ 1, 2k), k2arerealizable by connected graphs.

Theorem (Kinnerley, 2014?)

No pair of the form (r, r 1) can be realized by a tree.

Domination Game

Game on spanning trees
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For any integer 1, there exists a graph G and its spanning treeT such thatg(G) g(T) .

Domination Game

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G4:

Domination Game

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G4:

Tk: in Gkremove all but the middle vertical edges.

Domination Game


( )
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G4:

Tk: in Gkremove all but the middle vertical edges.

g(Gk) 52

k 1 and g(Tk) 2k+ 3.

Domination Game

Game on spanning subgraphs
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For any m3 there exists a 3-connected graph Gm and its

2-connected spanning subgraph Hm such thatg(Gm) 2m 2andg(Hm) =m.

Domination Game

Game on spanning subgraphs contd
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K8

K6 K6 K6 K6

x1 y1 x2 y2 x3 y3 x4 y4

a1,

1

a1,2

a1,3

a1,4

G4:

Hm is obtained from Gm by removing all the edges ai,jaj+1,i.

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Open problems

Domination Game

3/5-conjectures
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Conjecture (Kinnersley, West, Zamani, 2013?)For an n-vertex forest T without isolated vertices,

g(T)3n

5 and g(T)

3n+ 2

5 .

Domination Game

3/5-conjectures
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Conjecture (Kinnersley, West, Zamani, 2013?)For an n-vertex forest T without isolated vertices,

g(T)3n

5 and g(T)

3n+ 2

5 .

Conjecture (Kinnersley, West, Zamani, 2013?)

For an n-vertex connected graph G,

g(G)3n

5 and g(G)

3n+ 2

5 .

Domination Game

3/5-trees on 20 vertices
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Domination Game

3/5-trees on 20 vertices contd
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Domination Game

3/5-conjectures cont
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Theorem (Bujtas, 2014?)

The3/5-conjecture holds true for forests in which no two leaves

are at distance 4.

Domination Game

Computational complexity
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Problem

What is the computational complexity of the game dominationnumber?

Domination Game

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Problem


Problem

What is the computational complexity of the game dominationnumber on trees?

Domination Game

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Problem


Problem

What is the computational complexity of the game dominationnumber on trees?

Problem

Can we sayanything about the computational complexity of the

domination game?

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Game Over!

Domination Game
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Documents

Domination Game