The eternal domination number 
is the smallest cardinal
number of an eternal dominating set
of a graph 
in the one-guard model. It is the least number of guards that
can defend every sequence of attacks on unoccupied vertices.
Initially the occupied vertices form a dominating
set. At each attack, exactly one guard moves along a graph
edge to the attacked graph vertex, and the occupied
vertices must again form a dominating
set.
If 
is the domination number and 
is the clique
covering number, then
 |
The upper bound follows by keeping one guard in each clique of a clique partition.
The gamma-theta conjecture asserted that
implies 
.
Adamczewski and Klostermeyer (2026) disproved it using the graph
complement of the Berlekamp-van
Lint-Seidel graph, which has domination number
and eternal domination number both equal to 3 but clique
covering number greater than 3.
