Linear‐Time Algorithms for Scattering Number and Hamilton‐Connectivity of Interval Graphs

We prove that for all k ≤ − 1 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.