The general concept of the block-transformation graph Gαβγ was introduced in . 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 G101 are planar, outerplanar or minimally nonouterplanar. Further we establish a necessary and sufficient condition for the block-transformation graph G101 to have crossing number one.
Bulletin of Mathematical Sciences and Applications (Volume 10)
B. Basavanagoud and J. B. Veeragoudar, "A Criterion for (Non-)Planarity of The Block-Transformation Graph Gαβγ when αβγ = 101", Bulletin of Mathematical Sciences and Applications, Vol. 10, pp. 38-47, 2014