Geodesic
Splitting Cubic Circle Graphs
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 723-741

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We show that every 3-regular circle graph has at least two pairs of twin vertices; consequently no such graph is prime with respect to the split decomposition. We also deduce that up to isomorphism, K4 and K3,3 are the only 3-connected, 3-regular circle graphs.
Keywords: circle graph, split decomposition, regular graph 
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     author = {Traldi, Lorenzo},
     title = {Splitting {Cubic} {Circle} {Graphs}},
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     url = {http://geodesic.mathdoc.fr/item/DMGT_2016_36_3_a15/}
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Traldi, Lorenzo. Splitting Cubic Circle Graphs. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 723-741. http://geodesic.mathdoc.fr/item/DMGT_2016_36_3_a15/