Kayak Paddle Graph
A kayak paddle graph 
is the graph obtained by joining cycle graphs 
and 
by a path of length 
(Gallian 2018).
is isomorphic to the 3-barbell graph.
Kayak paddle graphs are planar, cactus, unit-distance and matchstick graphs. They are also bridged and traceable and have arboricity of 2.
Litersky (2011) proved that kayak paddle graphs are graceful when:
1. 
, 
,
2. 
(mod 4) for 
,
3. 
, 
(Litersky 2011, Gallian 2018).
REFERENCES
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.
Litersky, A. "Graceful Kayak Paddles." M.S. Thesis. Duluth, MN: University of Minnesota Duluth, 2011.
