B. Basavanagoud, Jaishri B. Veeragoudar, A Criterion for (Non-)Planarity of The Block-Transformation Graph Gαβγ when αβγ = 101, BMSA Volume 10, Bulletin of Mathematical Sciences and Applications (Volume 10)
https://www.scipress.com/BMSA.10.38
Abstract:
    The general concept of the block-transformation graph G<sup>αβγ</sup> was introduced in [1]. The vertices and blocks of a graph are its members. The block-transformation graph G101 of a graph G is the graph, whose vertex set is the union of vertices and blocks of G, in which two vertices are adjacent whenever the corresponding vertices of G are adjacent or the corresponding blocks of G are nonadjacent or the corresponding members of G are incident. In this paper, we present characterizations of graphs whose block-transformation graphs G<sup>101</sup> are planar, outerplanar or minimally nonouterplanar. Further we establish a necessary and sufficient condition for the block-transformation graph G<sup>101</sup> to have crossing number one.
Keywords:
    Crossing Number, Minimally Nonouterplanar, Outerplanar, Planar