Geodesic
The crossing numbers of join products of paths with graphs of order four
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 2, pp. 321-331

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Kulli and Muddebihal [V.R. Kulli, M.H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001) 87-97] gave the characterization of all pairs of graphs which join product is planar graph. The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. There are only few results concerning crossing numbers of graphs obtained as join product of two graphs. In the paper, the exact values of crossing numbers for join of paths with all graphs of order four, as well as for join of all graphs of order four with n isolated vertices are given.
Keywords: graph, drawing, path, crossing number, join product 
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Klešč, Marián; Schrötter, Stefan. The crossing numbers of join products of paths with graphs of order four. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 2, pp. 321-331. http://geodesic.mathdoc.fr/item/DMGT_2011_31_2_a8/