Linear-time algorithms for scattering number and hamilton-connectivity of interval graphs

We prove that for all integers k<0 an interval graph is -(k+1)-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. We also give an O(n+m) time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best-known O(n^3) time bound for solving this problem. As a consequence of our two results, the maximum k for which an interval graph is k-Hamilton-connected can be computed in O(n+m) time.