Adversarial Coloring, Covering
and Domination
Chip Klostermeyer
Dominating Set
γ=2
Independent Set
β=3
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
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
Basic Bounds
γ ≤ β ≤ γ∞ ≤ Θ
* One guard can defend a clique.
* Attacks on an independent set of size k
require k different guards
Upper Bound
Klostermeyer and MacGillivray proved
γ∞ ≤ C(β+1, 2)
C(n, 2) denotes binomial coefficient
Proof is algorithmic.
Lower Bound
• Upper bound:
γ∞ ≤ C(β+1, 2)
• 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 ?
None that are triangle free;
none with maximum-degree three.
Is there a triangle-free graph G with β = γ∞ < Θ ?
M-Eternal Dominating Sets
(all guards move)
γ ≤ γ ∞m ≤ β
3 by n grids: 4n/5+1 or 4n/5+2 guards needed
2 by 3 grid: 2 guards suffice
Protecting Edges
Attacks edges: guard must cross attacked edge. All guards
move. Guards must induce a VERTEX COVER
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.
Which graphs have α∞ = γ∞m ??
We characterize trees.
No graph with δ ≥ 2 except C4
Vertex Cover
• γ∞m < α∞ for all graphs of minimum degree
2, except for C4.
• γ∞m < α 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)
Attack vertex with guard, moves to empty neighbor
• 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?
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
(change to yellow)
Choose this vertex
change 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
Can we always start with a Х coloring?