A kayak paddle graph 
is the graph obtained by joining cycle graphs 
and 
by a path of length 
(Gallian 2018). A number of kayak paddle graphs are illustrated
above.
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).
See also
Barbell Graph, Cycle Graph, Lollipop Graph, Pan Graph, Tadpole Graph
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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.
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Cite this as:
Weisstein, Eric W. "Kayak Paddle Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KayakPaddleGraph.html