Linear‐Time Algorithms for Scattering Number and Hamilton‐Connectivity of Interval Graphs
Hajo J. Broersma, Jiřı́ Fiala, Petr A. Golovach, Tomáš Kaiser, Daniël Paulusma, Andrzej Proskurowski · Journal of Graph Theory · 2014
Abstract We prove that for all an interval graph is ‐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 time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best‐known 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 time.