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.

Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin.DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Harary,
F. Graph
Theory. Reading, MA: Addison-Wesley, 1994.Sloane, N. J. A.
Sequences A001429/M1438 and A236570
in "The On-Line Encyclopedia of Integer Sequences."Truszczyński,
M. "Graceful Unicyclic Graphs." Demonstatio Math.17, 377-387,
1984.