The 
-tadpole graph, also called a dragon
graph (Truszczyński 1984) or kite graph (Kim and Park 2006), is the graph obtained
by joining a cycle graph 
to a path graph 
with a bridge.
Special cases are summarized in the following table (where the names paw graph and banner graph appear in ISGCI).
 -tadpole graph | name |
 | paw graph |
 | hammer graph |
 | banner graph |
 |  -lollipop graph |
 |  -pan graph |
Precomputed properties of tadpole graphs are available in the Wolfram Language as GraphData[
"Tadpole", 
m, n
].
Koh et al. (1980) showed that 
-tadpole graphs are graceful
for 
,
1, or 3 (mod 4) and conjectured that all tadpole graphs are graceful
(Gallian 2018). Guo (1994) apparently completed the proof by filling in the missing
case in the process of showing that tadpoles are graceful when 
or 2 (mod 4) (Gallian 2018).
." J. Inner Mongolia Normal Univ., 24-29,
1994.ISGCI: Information System on Graph Class Inclusions v2.0. "List
of Small Graphs."