A unicyclic graph is a connected graph containing exactly one cycle (Harary 1994, p. 41). A connected unicyclic graph is therefore a pseudotree that is not a tree.
Truszczyński (1984) conjectured that all unicyclic graphs except the cycle graph 
with 
or 2 (mod 4) are graceful (Gallian 2018).
The numbers of unicyclic graphs on 
, 2, ... vertices are 0, 0, 1, 3, 9, 25, 68, 185, ... (OEIS
A236570), the first few of which are illustrated
above.
The corresponding numbers of connected unicyclic graphs are 0, 0, 1, 2, 5, 13, 33, 89, ... (OEIS A001429), the first few of which are illustrated above.
Examples of unicyclic classes of graphs include 
-caveman graphs, cycle
graphs 
,
pan graphs, sunlet graphs 
, and tadpole
graphs.
