Adversarial Coloring, Covering
and Domination
Chip Klostermeyer
Dominating Set
γ=2
Independent Set
β=3
Clique Cover
Θ=2
Graph
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
Eternal Dominating
Set
γ∞=3
γ=2
Attacked Vertex in red
Guards 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
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 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.
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?
Eviction Model – One Guard Moves
e∞=2
γ=2
Attacked Vertex in red
Attacked guard must have empty neighbor
Eviction: 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 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 needed
for 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