The paper doi:10.7151/dmgt.1639On Properties of Maximal 1-planar Graphs by Dávid Hudák, Tomáš Madaras and Yusuke Suzuki describes the construction of maximal 1-planar graphs with minimum degree 2 and with (relatively) small number of edges, namely, (8/3)*(n-2) (which is far less than the number of edges of a maximal n-vertex planar graph).