Computing Minimum Cut Sets for Circular-Arc Graphs| a New Approach

Let A be a set of n arcs on the unit circle. We present a new simple Theta(n log n)-time algorithm for computing a minimum cut set for A (and a maximum independent subset of A). Our solution is based on a dynamic maintenance scheme for a set S of n intervals on the line, that enables us to update the current minimum cut set for S, following an insertion or a deletion of an interval, in time O(c log n), where c is the size of the current minimum cut set. 1 Introduction Let A = fa 1 = [l 1 ; r 1 ]; : : : ; a n = [l n ; r n ]g be a set of n arcs on the unit circle, and consider the implicit circular-arc graph G(A) = (V(A); E(A)) defined as follows: Associate a node A i with each arc a i 2 A, and draw an edge between two nodes A i and A j if and only if a i " a j 6= ;. An independent subset of A is a set of pairwise non-intersecting arcs of A. Let b(A) be the size of a maximum independent subset of A. That is, any independent subset of A is of size at most b(A), and there exists a...