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Realizations of the Game Domination Number Bo stjan Bre sar, Sandi Klav zar, Ga sper Ko smrlj, Doug Rall Faculty of Mathematics and Physics University of Ljubljana, Slovenia 20th September 2012 Domination Game game on a finite

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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

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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 (γg) and Staller-start game (γ′

g)

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Basic problems

1 For a given graph G find γg(G) and/or γ′ g(G). 2 For a given pair (k, l) ∈ N × N find a graph G for which γg(G) = k

and γ′

g(G) = l.

Examples: γg(P) = 5, γ′

g(P) = 4, P Petersen graph

γg(Pn) = ⌈ n

2⌉

Kn realizes (1, 1) for any n ≥ 1 Km,n realizes (3, 2) for any m, n ≥ 1

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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 γg(G|A) ≤ γg(G|B) and γ′

g(G|A) ≤ γ′ g(G|B).

Corollary: |γg(G) − γ′

g(G)| ≤ 1.

Proof: u ∈ V (G), γg(G) ≤ γ′

g(G|N[u]) + 1 CP

≤ γ′

g(G) + 1

γ′

g(G) =

max

x∈V (G) γg(G|N[x]) + 1 CP

≤ γg(G) + 1

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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 {Gk; k ≥ 1} that realizes pair p. Theorem: For every n ≥ 1, γg(G ◦ Kn) = γg(G) and γ′

g(G ◦ Kn) = 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.

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Pair (k, k + 1)

1-connected prime: Tk k − 1

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Pair (k, k + 1)

1-connected prime: Tk k − 1 2-connected prime: Gk C2k

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Pair (k, k)

1-connected prime: P2k [K, W, Z]

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Pair (k, k)

1-connected prime: P2k [K, W, Z] 2-connected prime: C ′

k

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Pair (k, k)

1-connected prime: P2k [K, W, Z] 2-connected prime: C ′

k

C2k

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Pair (k, k)

1-connected prime: P2k [K, W, Z] 2-connected prime: C ′

k

C ′

k

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Pair (2k + 1, 2k)

1-connected prime: CCk C4k+2

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Pair (2k + 1, 2k)

1-connected prime: CCk C4k+2 2-connected prime: C4k+2 [K, W, Z]

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Pair (2k + 2, 2k + 1)

(2, 1) not realizable 1-connected prime: a very complicated one [Z]

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Pair (2k + 2, 2k + 1)

(2, 1) not realizable 1-connected prime: a very complicated one [Z] 2-connected prime: BLk C4k+2 P2P4

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Extremal realizations

3/5-conjecture: If G is an isolate-free graph of order n, then γg(G) ≤ 3n/5. Is the bound tight?

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Extremal realizations

3/5-conjecture: If G is an isolate-free graph of order n, then γg(G) ≤ 3n/5. Is the bound tight? yes ai ai F (fork) P5

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Construction using paths and forks

Path of order k ≥ 6: s s′ t′ t

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Construction using paths and forks

Choose one of the middle vertices: s s′ t′ t x

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Construction using paths and forks

Label the rest of the vertices: s s′ t′ t x u1 u2 uk−5

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Construction using paths and forks

Identify ui with ai ∈ V (Ti), where Ti ∈ {P5, F} for i = 1, 2, . . . , k − 5: s s′ t′ t x u1 u2 uk−5 T1 T2 Tk−5 Notation: T ℓ

k[T1, T2, . . . , Tk−5] where ℓ = d(s, x) + 1

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Construction using paths and forks

s s′ t′ t x u1 u2 uk−5 T1 T2 Tk−5 Notation: T ℓ

k[T1, T2, . . . , Tk−5] where ℓ = d(s, x) + 1

Theorem: T ℓ

k[T1, T2, . . . , Tk−5] is a 3/5-tree.

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Example

T 3

7 [P5, F]:

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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 γg(T|x) = γg(T) 3 γ′ g(T) = γg(T)

Theorem: (T ℓ

k[T1, . . . , Tk−5], x) is attachable.

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Another construction

G graph with V (G) = {v1, . . . , vn} and Hi, 1 ≤ i ≤ n, be a connected graph of order mi ≥ 2, and xi ∈ V (Hi). We denote by G[H1[x1], H2[x2], . . . , Hn[xn]] the graph of order n

i=1 mi formed by

identifying xi and vi for 1 ≤ i ≤ n. Theorem: (T1, x1) and (T2, x2) attachable 3/5-trees, then K2[T1[x1], T2[x2]] is a 3/5-tree.

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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, (Ti, xi) special attachable 3/5-trees for i = 1, . . . , n, then G[T1[x1], . . . , Tn[xn]] is a 3/5-graph.

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Examples

Special attachable 3/5-trees: P5, F , T 3

7 [P5, P5], T 4 7 [P5, P5]

(Question: Are all T ℓ

k-like trees special?)

K2[T 4

7 [P5, P5], P5] and K1,3[P5, P5, P5, P5]:

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Thank you for your attention! Questions?

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