We study four classical graph problems -- Hamiltonian path, Traveling salesman, Minimum spanning tree, and Minimum perfect matching on geometric graphs induced by bichromatic (red and blue) points. These problems have been widely studied for points in the Euclidean plane, and many of them are NP-hard. In this work, we consider these problems in two restricted settings: (i) collinear points and (ii) equidistant points on a circle. We show that almost all of these problems can be solved in linear time in these constrained, yet non-trivial settings.
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