Minimum Cuts for Circular-Arc Graphs

The problem of finding a minimum cut of n arcs on a unit circle is considered. It is shown that this problem can be solved in $\Theta (n \log n)$ time, which is optimal to within a constant factor. If the endpoints of the arcs are sorted, the problem can be solved in linear time. The solution to the minimum cut problem can be used to solve a minimum new facility problem in competitive location and a minimum partition set problem for the intersection model of a circle graph. As a by-product it is also shown that the maximum independent set of n arcs can be obtained in linear time, assuming the endpoints are sorted, which is much simpler than the most recent result of Masuda and Nakajima [SIAMI. Comput., 17 (1988), pp. 41–52].