A graph is $2$-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal $2$-planar if no edge can be added such that the resulting graph remains $2$-planar. A $2$-planar graph on $n$ vertices has at most $5n-10$ edges, and some (maximal) $2$-planar graphs -- referred to as optimal $2$-planar -- achieve this bound. However, in strong contrast to maximal planar graphs, a maximal $2$-planar graph may have fewer than the maximum possible number of edges. In this paper, we determine the minimum edge density of maximal $2$-planar graphs by proving that every maximal $2$-planar graph on $n\ge 5$ vertices has at least $2n$ edges. We also show that this bound is tight, up to an additive constant. The lower bound is based on an analysis of the degree distribution in specific classes of drawings of the graph. The upper bound construction is verified by carefully exploring the space of admissible drawings using computer support.

翻译：一个图被称为2-平面图，若其局部交叉数为2，即该图可画在平面上使得每条边至多有两个交叉。一个图被称为最大2-平面图，若添加任意一条边后所得图不再保持2-平面性。具有n个顶点的2-平面图至多有5n-10条边，且某些（最大）2-平面图（称为最优2-平面图）可达到这一上界。然而，与最大平面图形成鲜明对比的是，最大2-平面图的边数可能小于可能的最大边数。本文通过证明每个顶点数n≥5的最大2-平面图至少包含2n条边，确定了最大2-平面图的最小边密度。我们还证明该下界在加法常数意义下是紧的。下界的证明基于对图特定画法类别中顶点度分布的分析。上界构造则通过借助计算机辅助对可容许画法空间进行细致探索而得到验证。