ETERNAL DOMINATION
Chip Klostermeyer
6 vertices
7 edges
Dominating Set
γ=2
Graph
6 vertices
7 edges
Independent Set
β=3
Graph
6 vertices
10 edges
Clique Cover
Θ=2
Graph
Eternal Dominating Set
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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
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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
γ=2
Attacked Vertex in red
Guards on black vertices
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?
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
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.
Question: Is there a smaller one?
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.
Lower Bound
Upper bound:
γ∞ ≤ C(β+1, 2)
Certain large complements of Kneser graphs
require this many guards.
Problem: find small circulants where bound is
tight.
C22[1,2,4,5,9,11]
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.
Planar?
Is there a triangle-free graph G with β = γ∞ < Θ ?
Is γ∞(G x H) ≥ γ∞ (G) γ∞ (H)?
The Fundamental Conjecture
For any vertex v in any minimum eternal dominating
set D there is a vertex u adjacent to v such that
D–v+u
is an eternal dominating set.
Fundamental Theorem
Given any graph G and minimum eternal
dominating set D containing v, there is a minimum
eternal dominating set D’ not containing v.
Corollary: For all graphs G
γ∞(G-v) ≤ γ∞(G)
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
3 by n grids: about 4n/5 guards suffice for n ≥ 9
2 by 3 grid: γ∞m = 2
Conjecture: # guards for n by n grid = γ + O(1)
M-Eternal Dominating Sets
Known that γ∞m ≤ n/2; sharp for odd length paths, many trees
What about graphs with minimum degree 3?
Petersen graph is 2n/5; we know no other examples
with more than 3n/8 (and no large cubic ones with 3n/8)
Cubic Bipartite graphs: γ∞m ≤ 7n/16 [HKM]
• Improve upper bound for minimum degree three
• Find infinite families needing close to 2n/5 guards.
Proof idea
Cubic Bipartite graphs: γ∞m ≤ 7n/16
Remove perfect matching M. Cycles remain:
Long cycles adjacent to no 4-cycle (via M)
n/3 guards
Long cycles connected to 4-cycles (via M)
7n/16 guards (8-cycles are obstacle)
4-cycles connected to each other (via M)
3n/7 guards
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: Find graphs with β = 3 and e∞ = 5
• 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
• e m∞ ≤ β
• Grids: m by n solved for m ≤ 4
Bound: em∞ ≤ (n+2)(m+3)/5 – 4
for m, n ≥ 8
• Question: Is em∞ ≤ γ∞m for all G?
(swap model only, else star is counterexample)
Mixed Model
Combine eternal domination and eviction:
Attack at vertex w/o guard: guard moves there
Attack at vertex w/ guard : guard moves away
• Denote by m∞
• Question: Is m∞ ≤ 6 when β = 3?
• Question: Is m∞ = γ∞ for all G?
Eternal Independent Sets
• One model defined by Hartnell and Mynhardt
• Caro & Klostermeyer define alternate model:
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Maintain an independent set of guards eternally
Attacks are at vertices with guards (like eviction)
Maximize # of guards
One guard moves or all-guards move or ALL
guards move
Eternal Independent Sets
• Questions
• Find graphs where eternal independence # (all guards
move) equals size of maximum matching. It is true for
bipartite graphs.
• Find graphs where eternal independence # (all guards
move) equals the independence number
• Characterize graphs where eternal independence # (one
guard moves) equals size of maximum induced
matching (a lower bound for eternal independence #)
Protecting Edges
Attack edges, guard must cross edge. All guards
move, must induce VERTEX COVER.
α=3
Protecting Edges
α∞ = 3
Edge Protection
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Theorem: α ≤ α∞ ≤ 2α
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
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Which graphs have α∞ = γ∞m ?
Trees with property characterized.
No bipartite graph with δ ≥ 2 except C4
No graph with δ ≥ 2 except C4
Which graphs with pendant vertices?
Vertex Cover
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m-eternal domination number is less than
eternal vertex cover number for all graphs of
minimum degree 2, except for C4.
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m-eternal domination number is less than
vertex cover number for all graphs of minimum
degree 2 and girth 7 and ≥ 9.
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What about girths 5, 6, 8?