ETERNAL DOMINATION

Chip Klostermeyer

Graph

6 vertices7 edges

Dominating Set γ=2

Graph

6 vertices7 edges

Independent Set β=3

Graph

6 vertices10 edges

Clique Cover Θ=2

Eternal Dominating Set

• Defend graph against sequence of attacks at vertices

• At most one guard per vertex• Send guard to attacked vertex• Guards must induce dominating set• One guard moves at a time

(later, we allow all guards to move)

2-player game

• Attacker chooses vertex with no guard to attack

• Defender chooses guard to send to attacked vertex (must be sent from neighboring vertex)

• Attacker wins if after some # of attacks, guards do not induce dominating set

• Defender wins otherwise

Attacked Vertex in redGuards on black vertices

Eternal Dominating Set γ∞=3 γ γ = 2

Second attack at red vertex forces guards to not be a dominating set.

3 guards needed

Eternal Dominating Set γ∞=3 γ = 2

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3 guards needed

Eternal Dominating Set γ∞=3 γ = 2

Basic Bounds

γ ≤ β ≤ γ∞ ≤ Θ

Because one guard can defend a clique and

attacks on an independent set of size k require

k different guards

Problem

Goddard, Hedetniemi, Hedetniemi asked if

γ∞ ≤ c * β

and they showed graphs for which

γ∞ < Θ

Smallest known has 11 vertices.

Question: Is there a smaller one?

Upper Bound

Klostermeyer and MacGillivray proved

γ∞ ≤ C(β+1, 2)

C(n, 2) denotes binomial coefficient

Proof is algorithmic.

Proof idea

Guards located on independent sets of size 1, 2, …,β

Defend with guard from smallest set possible

Proof idea

Guards located on independent sets of size 1, 2, …,β

Swapping guard with attacked vertex destroys independence! Solution….

Proof idea

Guards located on independent sets of size 1, 2, …,β

Choose union of independent sets to be LARGE as possible

Proof idea

Guards located on independent sets of size 1, 2, …,β

After yellow guard moves, we have all our independent sets.

Lower Bound

Upper bound:

γ∞ ≤ C(β+1, 2)

Certain large complements of Kneser graphs require this many guards.

Problem: find small circulants where bound is tight.

C22[1,2,4,5,9,11]

γ ≤ β ≤ γ∞ ≤ Θ

γ∞ =Θ for

Perfect graphs [follows from PGT]Series-parallel graphs [Anderson et al.]Powers of Cycles and their complements [KM]Circular-arc graphs [Regan]

Open problem: planar graphs

Open Questions

Is there a graph G with γ = γ∞ < Θ ?

No triangle-free; none with maximum-degree three.

Planar?

Is there a triangle-free graph G with β = γ∞ < Θ ?

Is γ∞(G x H) ≥ γ∞ (G) γ∞ (H)?

The Fundamental Conjecture

For any vertex v in any minimum eternal dominating set D there is a vertex u adjacent to v such that

D – v + u

is an eternal dominating set.

Fundamental Theorem

Given any graph G and minimum eternal dominating set D containing v, there is a minimum eternal dominating set D’ not containing v.

Corollary: For all graphs G

γ∞(G-v) ≤ γ∞(G)

M-Eternal Dominating Set γ∞

m=2

All guards can move in response to attack

M-Eternal Dominating Sets

γ ≤ γ∞m ≤ β

Exact bounds known for trees, 2 by n, 4 by n grids

3 by n grids: about 4n/5 guards suffice for n ≥ 9

2 by 3 grid: γ∞m = 2

Conjecture: # guards for n by n grid = γ + O(1)

M-Eternal Dominating Sets

Known that γ∞

m ≤ n/2; sharp for odd length paths, many trees

What about graphs with minimum degree 3?

Petersen graph is 2n/5; we know no other examples with more than 3n/8 (and no large cubic ones with 3n/8)

Cubic Bipartite graphs: γ∞m ≤ 7n/16 [HKM]

• Improve upper bound for minimum degree three• Find infinite families needing close to 2n/5 guards.

Proof idea

Cubic Bipartite graphs: γ∞m ≤ 7n/16

Remove perfect matching M. Cycles remain:

Long cycles adjacent to no 4-cycle (via M)

n/3 guards

Long cycles connected to 4-cycles (via M)

7n/16 guards (8-cycles are obstacle)

4-cycles connected to each other (via M)

3n/7 guards

Attacked Vertex in red

Attacked guard must have empty neighbor

e∞=2 γ = 2

Eviction Model: One Guard Moves

•e∞ ≤ Θ

• e∞ ≤ β for bipartite graphs

• e∞ > β for some graphs

• e∞ ≤ β when β=2

• e∞ ≤ 5 when β = 3

•Question: Find graphs with β = 3 and e∞ = 5

•Question: Is e∞ ≤ γ∞ for all G?

Eviction: One guard moves

Eviction Model: All Guards Move

e∞m = 2

Attacked vertex must remain empty for one time period

• em∞ ≤ β

• Grids: m by n solved for m ≤ 4 Bound: em

∞ ≤ (n+2)(m+3)/5 – 4 for m, n ≥ 8

• Question: Is em∞ ≤ γ∞

m for all G?

(swap model only, else star is counterexample)

Eviction: All guards move

Combine eternal domination and eviction: Attack at vertex w/o guard: guard moves there Attack at vertex w/ guard : guard moves away

•Denote by m∞

•Question: Is m∞ ≤ 6 when β = 3?

•Question: Is m∞ = γ∞ for all G?

Mixed Model

Eternal Independent Sets

• One model defined by Hartnell and Mynhardt

• Caro & Klostermeyer define alternate model:

• Maintain an independent set of guards eternally• Attacks are at vertices with guards (like eviction)• Maximize # of guards• One guard moves or all-guards move or ALL

guards move

Eternal Independent Sets

• Questions

• Find graphs where eternal independence # (all guards move) equals size of maximum matching. It is true for bipartite graphs.

• Find graphs where eternal independence # (all guards move) equals the independence number

• Characterize graphs where eternal independence # (one guard moves) equals size of maximum induced matching (a lower bound for eternal independence #)

Protecting Edges

Attack edges, guard must cross edge. All guards move, must induce VERTEX COVER.

α = 3

Protecting Edges

α∞ = 3

• Theorem: α ≤ α∞ ≤ 2α• Which graphs have α = α∞? Grids Kn X G Circulants, others.

Is it true for vertex-transitive graphs? Is it true for G X H if it is true for G and/or H?

Edge Protection

More Edge Protection

• Which graphs have α∞ = γ∞m ?

• Trees with property characterized.• No bipartite graph with δ ≥ 2 except C4

• No graph with δ ≥ 2 except C4

• Which graphs with pendant vertices?

Vertex Cover

• m-eternal domination number is less than eternal vertex cover number for all graphs of minimum degree 2, except for C4.

• m-eternal domination number is less than vertex cover number for all graphs of minimum degree 2 and girth 7 and ≥ 9.

• What about girths 5, 6, 8?