Adversarial Coloring, Covering and Domination

Chip Klostermeyer

Dominating Set γ=2

Independent Set β=3

Graph

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=2

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

3 guards needed

Eternal Dominating Set γ∞=3 γ=2

?

?

3 guards needed

Eternal Dominating Set γ∞=3 γ=2

Applications

Military Defense (original problem dates to Emperor Constantine)

Autonomous Systems (foolproof model)

File Migration

File Migration for server maintenance (eviction model)

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)

Upper Bound

Klostermeyer and MacGillivray proved

γ∞ ≤ C(β+1, 2)

C(n, 2) denotes binomial coefficient

Proof is algorithmic.

Proof ideaGuards located on independent sets of size 1, 2, …,β

Defend with guard from smallest set possible

Proof ideaGuards located on independent sets of size 1, 2, …,β

Swapping guard with attacked vertex destroys independence!! Solution….

Proof ideaGuards located on independent sets of size 1, 2, …,β

Choose union of independent sets to be LARGE as possible

Proof ideaGuards located on independent sets of size 1, 2, …,β

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

Key points in proof

• Independent sets induce a dominating set since independent set of size β is a dominating set.

• Can show that even if guard moves from the independent set of size β, after move there will still be an independent set of size β.

Lower Bound?

• Upper bound:

γ∞ ≤ C(β+1, 2)

• But is it tight?

• Yes. Goldwasser and Klostermeyer proved that certain (large) complements of Kneser graphs require this many guards.

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.

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

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 (latter by Finbow et al.)

3 by n grids: ≤ 8n/9 guards needed (improved by Finbow, Messiginer et al).

2 by 3 grid: 2 guards suffice

Conjecture: # guards needed in n by n grid is γ + O(1)

Eternal Total Domination

• Require dominating set to be total at all times.

• Example: 4 guards (if one moves at a time). 3 guards (if all can move)

Guards move up and down in tandem

Eternal Total Domination

γ∞ < γ∞t ≤ γ∞ + γ ≤ 2Θ

γ ≤ γt ≤ γ∞tm ≤ 2Θ-1

We characterize the graphs where the last inequality is tight.

Exact bounds known for 2 by n and 3 by n grids.

Protecting Edges• Attacks on edges: guard must cross

attacked edge. All guards move.

• Guards must induce a VERTEX COVER

α = 3

Protecting Edges

α∞ = 3

Results

• α ≤ α∞ ≤ 2α

• Graphs achieving upper bound characterized [Klost.-Mynhardt]

• Trees require # internal vertices + 1

Edge Protection

• 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?

More Edge Protection

• Which graphs have α∞ = γ∞m ??

• We characterize which trees.

• No bipartite graph with δ ≥ 2 except C4

• No graph with δ ≥ 2 except C4

• Graphs with pendant vertices??

Explain criticality in edge protection!

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 5, 6, 8?

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: 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

Eviction: All guards move

• em∞ ≤ β

• Question: Is em∞ ≤ γ∞

m for all G?

Eternal Graph Coloring

Colors as frequencies in cellular network.

What if user wants to change frequencies for security?

Two player game:

Player 1 chooses proper coloring Player 2 chooses vertex whose color must change Player 1 must choose new color for that vertex etc.

How many colors ensure Player 1 always has a move?

Player 2 chooses this vertex (changes to yellow)

Choose this vertex changes to ?

Five colors neededfor Player 1 to win

Results

Χ∞ ≤ 2Х (tighter bound: 2Хc )

Χ∞ = 4 only for bipartite or odd cycles

Exists a planar graph with Χ∞ = 8

Δ+ 2 ≥ Χ∞ ≥ Х + 1

Χ∞(Wheel) = 6 [Note that deleting center vertex decrease Χ∞ by 2 here]

Brooks Conjectures:

Χ∞ = Х + 1 if and only if G is complete graph or odd cycle

Χ∞ = Δ + 2 (those with X = Δ, complete graphs, odd cycles, some complete multi-partites, others?)

Future work: For which graphs is Χ∞ = 5? Complexity of deciding that question