Chicken (game) respectively. The line in graph on the left shows the optimum probability of playing the escalated strategy for player Y as a function of "x". The line in the second graph shows the optimum probability of playing the escalated strategy for player X as a function of "y" (the axes have not been rotated, so the dependent variable is plotted on the abscissa, and the independent variable is plotted on the ordinate). The Nash equilibria are where the players' correspondences agree, i.e., cross. These are shown with points in the right hand graph. The best response mappings agree (i.e., cross)
Topological graph embeddings of graphs in surfaces (that is, drawings without crossings). A fundamental problem in extremal graph theory is the following: what is the maximum number of edges that a graph of "n" vertices can have if it contains no subgraph belonging to a given class of "forbidden subgraphs"? The prototype of such results is Turán's theorem, where there is one forbidden subgraph: a complete graph with "k" vertices ("k" is fixed). Analogous questions can be raised for topological and geometric graphs, with the difference that now certain "geometric subconfigurations" are forbidden. Historically, the first instance of such a theorem is