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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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].
Two players, Dominator (D) and Staller (S), take turns choosing a
vertex in G.
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].
Two players, Dominator (D) and Staller (S), take turns choosing a
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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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].
Two players, Dominator (D) and Staller (S), take turns choosing a
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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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)
d1, s1, d2, s2, . . . denotes the sequence of moves.
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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)
d1, s1, d2, s2, . . . denotes the sequence of moves.
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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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)
d1, s1, d2, s2, . . . denotes the sequence of moves.
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
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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
The game on P5 contd
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g 5
s1
The game on P5 contd
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g 5
s1d1
The game on P5 contd
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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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SupposeD and Splay Game 1 on G.
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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SupposeD and Splay Game 1 on G.
Let Cdenote the total set of vertices chosen by D and S.Then C is a dominating set ofG.
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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SupposeD and Splay Game 1 on G.
Let Cdenote the total set of vertices chosen by D and S.Then C is a dominating set ofG.
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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Theorem (Bresar, K., Rall, 2010; Kinnersley, West, Zamani, 2013?)
For any graph G, |g(G) g(G)| 1.
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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Theorem (Bresar, K., Rall, 2010; Kinnersley, West, Zamani, 2013?)
For any graph G, |g(G) g(G)| 1.
Apartially dominated graphis a graph in which some verticeshave already been dominated in some turns of the gamealready played.
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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Theorem (Bresar, K., Rall, 2010; Kinnersley, West, Zamani, 2013?)
For any graph G, |g(G) g(G)| 1.
Apartially dominated graphis a graph in which some verticeshave already been dominated in some turns of the gamealready played.
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).
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).
D will keepthe rule: the set of vertices dominated in Game Ais a superset of vertices dominated in Game B.
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).
D will keepthe rule: the set of vertices dominated in Game Ais a superset of vertices dominated in Game B.
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.
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).
D will keepthe rule: the set of vertices dominated in Game Ais a superset of vertices dominated in Game B.
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
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.
Bythe rule, Game A finishes no later than Game B.
Domination Game
Proof of Continuation Principle contd
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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.
Bythe rule, Game A finishes no later than 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
Proof of Continuation Principle contd
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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.
Bythe rule, Game A finishes no later than 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.
Domination Game
Proof of the theorem
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Consider Game 1 and let vbe the first move ofD.
Let G be the resulting partially dominated graph.
Domination Game
Proof of the theorem
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Consider Game 1 and let vbe the first move ofD.
Let G be the resulting partially dominated graph.
g(G) g(G) + 1.
Domination Game
Proof of the theorem
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Consider Game 1 and let vbe the first move ofD.
Let G be the resulting partially dominated graph.
g(G) g(G) + 1.
By Continuation Principle, g(G) g(G).
Domination Game
Proof of the theorem
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Consider Game 1 and let vbe the first move ofD.
Let G be the resulting partially dominated graph.
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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A pair (r, s) of integers is realizableif there exists a graph G such
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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A pair (r, s) of integers is realizableif there exists a graph G such
that g(G) =r and g(G) =s. By the theorem, only possiblerealizable pairs are: (r, r), (r, r+ 1), (r, r 1).
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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A pair (r, s) of integers is realizableif there exists a graph G such
that g(G) =r and g(G) =s. By the theorem, only possiblerealizable pairs are: (r, r), (r, r+ 1), (r, r 1).
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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Theorem (Bresar, K., Rall, 2013)
For any integer 1, there exists a graph G and its spanning treeT such thatg(G) g(T) .
Domination Game
Game on spanning trees
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Theorem (Bresar, K., Rall, 2013)
For any integer 1, there exists a graph G and its spanning treeT such thatg(G) g(T) .
G4:
Domination Game
Game on spanning trees
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Theorem (Bresar, K., Rall, 2013)
For any integer 1, there exists a graph G and its spanning treeT such thatg(G) g(T) .
G4:
Tk: in Gkremove all but the middle vertical edges.
Domination Game
Game on spanning trees
( )
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Theorem (Bresar, K., Rall, 2013)
For any integer 1, there exists a graph G and its spanning treeT such thatg(G) g(T) .
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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Theorem (Bresar, K., Rall, 2013)
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.
Domination Game
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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
Computational complexity
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Problem
What is the computational complexity of the game dominationnumber?
Problem
What is the computational complexity of the game dominationnumber on trees?
Domination Game
Computational complexity
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Problem
What is the computational complexity of the game dominationnumber?
Problem
What is the computational complexity of the game dominationnumber on trees?
Problem
Can we sayanything about the computational complexity of the
domination game?
Domination Game
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Game Over!
Domination Game
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