Biplanar graphs: A survey (Q1388965)

A biplanar graph is the union of two planar graphs; thus its thickness is at most two. In this survey article, the author presents various properties of biplanar graphs, two special families of biplanar graphs (doubly linear graphs and rectangle visibility graphs), and extensions to other surfaces, both orientable and nonorientable (biembeddings). \textit{A. Mansfield} [Math. Proc. Camb. Philos. Soc. 93, 9-23 (1983; Zbl 0503.68048)] showed that determining whether a given graph is biplanar is an NP-complete problem. However, as shown in the present paper, every graph is homeomorphic to a biplanar graph.