In 1975, Plesn\'ik characterized all triangle-free planar graphs as having a diameter $2$. We characterize all triangle-free projective-planar graphs having a diameter $2$ and discuss some applications. In particular, the main result is applied to calculate the analogue of clique numbers for graphs, namely, colored mixed graphs, having different types of arcs and edges.

翻译：1975年,Plesn\'ik将所有无三角平面图都定性为直径为$2的图。我们将直径为$2的所有无三角投影平面图定性为$2,并讨论一些应用。特别是,主要结果用于计算图表的圆形数字的类比,即彩色混合图,其弧形和边缘类型不同。