Realizations of the Game Domination Number Bo stjan Bre sar, Sandi - PowerPoint PPT Presentation

Realizations of the Game Domination Number Boˇ stjan Breˇ sar, Sandi Klavˇ zar, Gaˇ sper Koˇ smrlj, Doug Rall Faculty of Mathematics and Physics University of Ljubljana, Slovenia 20th September 2012

Domination Game game on a finite graph two players, Dominator (male) and Staller (female) legal move: set of dominated vertices enlarges by at least one Dominator-start game ( γ g ) and Staller-start game ( γ ′ g ) 20th September 2012 2 / 17

Basic problems 1 For a given graph G find γ g ( G ) and/or γ ′ g ( G ). 2 For a given pair ( k , l ) ∈ N × N find a graph G for which γ g ( G ) = k and γ ′ g ( G ) = l . Examples: γ g ( P ) = 5, γ ′ g ( P ) = 4, P Petersen graph γ g ( P n ) = ⌈ n 2 ⌉ K n realizes (1 , 1) for any n ≥ 1 K m , n realizes (3 , 2) for any m , n ≥ 1 20th September 2012 3 / 17

What pairs can be realizable? For a vertex subset S of a graph G , G | S denotes the partially dominated graph in which vertices from S are already dominated. Continuation Principle : [Kinnersley, West, Zemani] For any graph G and B ⊆ A ⊆ V ( G ) it follows that γ g ( G | A ) ≤ γ g ( G | B ) and γ ′ g ( G | A ) ≤ γ ′ g ( G | B ). Corollary : | γ g ( G ) − γ ′ g ( G ) | ≤ 1 . Proof : CP γ g ( G ) ≤ γ ′ ≤ γ ′ u ∈ V ( G ) , g ( G | N [ u ]) + 1 g ( G ) + 1 CP γ ′ g ( G ) = x ∈ V ( G ) γ g ( G | N [ x ]) + 1 max ≤ γ g ( G ) + 1 20th September 2012 4 / 17

Realizations and lexicographic product Problem : For every r ≥ 1 and pair p ∈ { ( k , k + 1) , ( k , k ) , ( k + 1 , k ) } find a family of r -connected graphs { G k ; k ≥ 1 } that realizes pair p . Theorem : For every n ≥ 1, γ g ( G ◦ K n ) = γ g ( G ) and γ ′ g ( G ◦ K n ) = γ ′ g ( G ). Recall that κ ( G ◦ H ) = κ ( G ) | V ( H ) | (if G not complete). Graph is prime if it is not constructed as in the above theorem. 20th September 2012 5 / 17

Pair ( k , k + 1) 1-connected prime: T k k − 1 20th September 2012 6 / 17

Pair ( k , k + 1) 1-connected prime: T k k − 1 2-connected prime: G k C 2 k 20th September 2012 6 / 17

Pair ( k , k ) 1-connected prime: P 2 k [K, W, Z] 20th September 2012 7 / 17

Pair ( k , k ) 1-connected prime: P 2 k [K, W, Z] 2-connected prime: C ′ k 20th September 2012 7 / 17

Pair ( k , k ) 1-connected prime: P 2 k [K, W, Z] 2-connected prime: C ′ k C 2 k 20th September 2012 7 / 17

Pair ( k , k ) 1-connected prime: P 2 k [K, W, Z] 2-connected prime: C ′ k C ′ k 20th September 2012 7 / 17

Pair (2 k + 1 , 2 k ) 1-connected prime: CC k C 4 k +2 20th September 2012 8 / 17

Pair (2 k + 1 , 2 k ) 1-connected prime: CC k C 4 k +2 2-connected prime: C 4 k +2 [K, W, Z] 20th September 2012 8 / 17

Pair (2 k + 2 , 2 k + 1) (2 , 1) not realizable 1-connected prime: a very complicated one [Z] 20th September 2012 9 / 17

Pair (2 k + 2 , 2 k + 1) (2 , 1) not realizable 1-connected prime: a very complicated one [Z] 2-connected prime: BL k C 4 k +2 P 2 � P 4 20th September 2012 9 / 17

Extremal realizations 3/5-conjecture : If G is an isolate-free graph of order n , then γ g ( G ) ≤ 3 n / 5. Is the bound tight? 20th September 2012 10 / 17

Extremal realizations 3/5-conjecture : If G is an isolate-free graph of order n , then γ g ( G ) ≤ 3 n / 5. Is the bound tight? yes F (fork) P 5 a i a i 20th September 2012 10 / 17

Construction using paths and forks Path of order k ≥ 6: s ′ t ′ s t 20th September 2012 11 / 17

Construction using paths and forks Choose one of the middle vertices: s ′ t ′ s x t 20th September 2012 11 / 17

Construction using paths and forks Label the rest of the vertices: u k − 5 u 1 u 2 s ′ t ′ s x t 20th September 2012 11 / 17

Construction using paths and forks Identify u i with a i ∈ V ( T i ), where T i ∈ { P 5 , F } for i = 1 , 2 , . . . , k − 5: u 1 u 2 u k − 5 s ′ t ′ s x t T k − 5 T 1 T 2 Notation: T ℓ k [ T 1 , T 2 , . . . , T k − 5 ] where ℓ = d ( s , x ) + 1 20th September 2012 11 / 17

Construction using paths and forks u k − 5 u 1 u 2 s ′ t ′ s x t T k − 5 T 1 T 2 Notation: T ℓ k [ T 1 , T 2 , . . . , T k − 5 ] where ℓ = d ( s , x ) + 1 Theorem : T ℓ k [ T 1 , T 2 , . . . , T k − 5 ] is a 3/5-tree. 20th September 2012 11 / 17

Example T 3 7 [ P 5 , F ]: 20th September 2012 12 / 17

Attachable trees T a tree and x ∈ V ( T ), then attachable tree is a pair ( T , x ) provided: 1 x is an optimal-start vertex for Dominator in Game 1 on T 2 γ g ( T | x ) = γ g ( T ) 3 γ ′ g ( T ) = γ g ( T ) Theorem : ( T ℓ k [ T 1 , . . . , T k − 5 ] , x ) is attachable. 20th September 2012 13 / 17

Another construction G graph with V ( G ) = { v 1 , . . . , v n } and H i , 1 ≤ i ≤ n , be a connected graph of order m i ≥ 2, and x i ∈ V ( H i ). We denote by G [ H 1 [ x 1 ] , H 2 [ x 2 ] , . . . , H n [ x n ]] the graph of order � n i =1 m i formed by identifying x i and v i for 1 ≤ i ≤ n . Theorem : ( T 1 , x 1 ) and ( T 2 , x 2 ) attachable 3/5-trees, then K 2 [ T 1 [ x 1 ] , T 2 [ x 2 ]] is a 3/5-tree. 20th September 2012 14 / 17

General case Attachable tree ( T , x ) is special if for any optimal first move of Staller in Game 2 that is different from x , Dominator can optimally reply with a move on x . Theorem : If G connected graph of order n , ( T i , x i ) special attachable 3/5-trees for i = 1 , . . . , n , then G [ T 1 [ x 1 ] , . . . , T n [ x n ]] is a 3/5-graph. 20th September 2012 15 / 17

Examples Special attachable 3/5-trees: P 5 , F , T 3 7 [ P 5 , P 5 ], T 4 7 [ P 5 , P 5 ] (Question: Are all T ℓ k -like trees special?) K 2 [ T 4 7 [ P 5 , P 5 ] , P 5 ] and K 1 , 3 [ P 5 , P 5 , P 5 , P 5 ]: 20th September 2012 16 / 17

Thank you for your attention! Questions? 20th September 2012 17 / 17