Geodesic
Non-1-Planarity of Lexicographic Products of Graphs
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 1103-1114

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In this paper, we show the non-1-planarity of the lexicographic product of a theta graph and K2. This result completes the proof of the conjecture that a graph G ◦ K2 is 1-planar if and only if G has no edge belonging to two cycles.
Keywords: 1-planar graph, lexicographic product 
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Matsumoto, Naoki; Suzuki, Yusuke. Non-1-Planarity of Lexicographic Products of Graphs. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 1103-1114. http://geodesic.mathdoc.fr/item/DMGT_2021_41_4_a15/