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Adversarial Coloring, Covering and Domination
Chip Klostermeyer
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Dominating Set γ=2
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Independent Set β=3
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Graph
Clique Cover Θ=2
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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)
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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
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Attacked Vertex in redGuards on black vertices
Eternal Dominating Set γ∞=3 γ=2
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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
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Applications
Military Defense (original problem dates to Emperor Constantine)
Autonomous Systems (foolproof model)
File Migration
File Migration for server maintenance (eviction model)
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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
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Problem
Goddard, Hedetniemi, Hedetniemi asked if
γ∞ ≤ c * β
And they showed graphs for which
γ∞ < Θ
(smallest known has 11 vertices)
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Upper Bound
Klostermeyer and MacGillivray proved
γ∞ ≤ C(β+1, 2)
C(n, 2) denotes binomial coefficient
Proof is algorithmic.
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Proof ideaGuards located on independent sets of size 1, 2, …,β
Defend with guard from smallest set possible
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Proof ideaGuards located on independent sets of size 1, 2, …,β
Swapping guard with attacked vertex destroys independence!! Solution….
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Proof ideaGuards located on independent sets of size 1, 2, …,β
Choose union of independent sets to be LARGE as possible
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Proof ideaGuards located on independent sets of size 1, 2, …,β
After yellow guard moves, we have all our independent sets.
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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.
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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.
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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
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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 ?
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M-Eternal Dominating Set γ∞
m=2
All guards can move in response to attack
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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)
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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
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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.
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Protecting Edges• Attacks on edges: guard must cross
attacked edge. All guards move.
• Guards must induce a VERTEX COVER
α = 3
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Protecting Edges
α∞ = 3
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Results
• α ≤ α∞ ≤ 2α
• Graphs achieving upper bound characterized [Klost.-Mynhardt]
• Trees require # internal vertices + 1
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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?
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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!
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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?
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Attacked Vertex in red
Attacked guard must have empty neighbor
e∞=2 γ=2
Eviction Model – One Guard Moves
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•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
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Eviction Model – All Guards Move
e∞m = 2
Attacked vertex must remain empty for one time period
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Eviction: All guards move
• em∞ ≤ β
• Question: Is em∞ ≤ γ∞
m for all G?
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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?
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Player 2 chooses this vertex (changes to yellow)
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Choose this vertex changes to ?
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Five colors neededfor Player 1 to win
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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]
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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