A criterion for (non-)planarity of theblock-transformation graph G αβγ when αβγ = 101

The general concept of the block-transformation graph G αβγ was introduced in (1). The vertices and blocks of a graph are its members. The block-transformation graph G 101 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 101 are planar, outerplanar or minimally nonouterplanar. Further we establish a necessary and sufficient condition for the block- transformation graph G 101 to have crossing number one.