1. Adversarial Coloring, Covering and Domination Chip Klostermeyer

2. Dominating Set γ=2

3. Independent Set β=3

4. Clique Cover Θ=2

5. 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

6. 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

7. Attacked Vertex in red Guards on black vertices Eternal Dominating Set γ ∞ =3 γ=2

8. Second attack at red vertex forces guards to not be a dominating set. 3 guards needed Eternal Dominating Set γ ∞ =3 γ=2 ? ?

9. 3 guards needed Eternal Dominating Set γ ∞ =3 γ=2

10. Basic Bounds γ ≤ β ≤ γ ∞ ≤ Θ * One guard can defend a clique. * Attacks on an independent set of size k require k different guards

11. Upper Bound Klostermeyer and MacGillivray proved γ ∞ ≤ C(β+1, 2) C(n, 2) denotes binomial coefficient Proof is algorithmic.

12. Lower Bound Upper bound: γ ∞ ≤ C(β+1, 2) Goldwasser and Klostermeyer proved that certain (large) complements of Kneser graphs require this many guards.

13. 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

14. 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 ?

15. 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

16. Protecting Edges α ∞ = 3 Attacks edges: guard must cross attacked edge. All guards move. Guards must induce a VERTEX COVER

17. Results α ≤ α∞ ≤ 2α Graphs achieving upper bound characterized [Klost.-Mynhardt] Trees require # internal vertices + 1

18. Edge Protection Which graphs have α = α∞? Grids K n X G Circulants, others. Which graphs have α∞ = γ ∞ m ?? We characterize trees. No graph with δ ≥ 2 except C 4

19. Vertex Cover γ ∞ m < α∞ for all graphs of minimum degree 2, except for C 4. γ ∞ m < α for all graphs of minimum degree 2 and girth 7 and ≥ 9. What about 5, 6, 8?

20. Attacked Vertex in red Attacked guard must have empty neighbor e ∞ =2 γ=2 Eviction Model – One Guard Moves

21. 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) Attack vertex with guard, moves to empty neighbor

22. 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?

23. Player 2 chooses this vertex (change to yellow)

24. Choose this vertex change to ?

25. Five colors needed for Player 1 to win

26. Results Χ ∞ ≤ 2Х (tighter bound: 2Х c ) 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]

27. 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?