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)
Graph homomorphism can be solved by brute-force in time |"V"("H")|, so polynomial in the size of the input graph "H". In other words, the problem is trivially in P for graphs "G" of bounded size. The interesting question is then what other properties of "G", beside size, make polynomial algorithms possible. The crucial property turns out to be treewidth, a measure of how tree-like the graph is. For a graph "G" of treewidth at most "k" and a graph "H", the homomorphism problem can be solved in time |"V"("H")| with a standard dynamic programming approach. In fact, it is enough to assume