On Plick Graphs with Coarseness Number One

The coarseness ξ ( G ) of a graph G is the maximum number of mutually line-disjoint nonplanar subgraphs of G . Clearly, ξ ( G ) = 1 if and only if G is nonplanar and G has no two line-disjoint subgraphs homeomorphic to K 3,3 or K 5 . In this paper, we obtain a necessary and sufficient condition for plick graph P n ( G ); n ≥ 1 to have coarseness number one.