Nonisomorphic graphs may have the same domination polynomial. A graph that does not share a domination
polynomial with any other nonisomorphic graph is said to be dominating unique
(or 
-unique
for short) (Akbari et al. 2010).
The numbers of dominating unique graphs on 
, 2, ... vertices are 1, 2, 4, 9, 21, 52, 168, 666, 3605,
27513, ... (OEIS A378516), the first few of
which are illustrated above. Classes of graphs that are dominating unique include
complete graph, cycle
graphs, empty graphs, hypercube
graphs, pan graphs, star
graphs, and wheel graphs.
Graphs that share the same domination polynomial are said to be dominating equivalent, dominating nonunique, or co-dominating graphs.
