An efficient dominating set of a simple graph 
is a subset 
of its vertex set such that
every graph vertex belongs to the closed
neighborhood of exactly one element of 
. Equivalently, the closed
neighborhoods centered at elements of 
partition the vertex set. Such
a set is also called a perfect 1-code in the graph
(Biggs 1973; Huang et al. 2018, p. 548).
An efficient dominating set is an independent vertex set and a dominating set. In a 
-regular graph with 
vertices, it has size 
,
since each of its closed neighborhoods has
size 
.
For example, the cycle graph 
has an efficient dominating set obtained by taking every
third graph vertex.
For an abelian group 
and an undirected Cayley graph
with connection set 
, a subset 
is an efficient dominating set iff
every element of 
has a unique expression 
with 
and 
(Huang et al. 2018; Hafidh et al. 2026,
p. 5).
