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

TL;DR

This paper provides a linear-time algorithm for computing the scattering number of interval graphs and characterizes their Hamilton-connectivity, extending previous results and enabling efficient analysis of their connectivity properties.

Contribution

It introduces a linear-time algorithm for the scattering number of interval graphs and completes the characterization of their Hamilton-connectivity.

Findings

Linear-time algorithm for scattering number computation

Complete characterization of Hamilton-connectivity in interval graphs

Improved time complexity from O(n^4) to O(m+n)

Abstract

Hung and Chang showed that for all k>=1 an interval graph has a path cover of size at most k if and only if its scattering number is at most k. They also showed that an interval graph has a Hamilton cycle if and only if its scattering number is at most 0. We complete this characterization by proving that for all k<=-1 an interval graph is -(k+1)-Hamilton-connected if and only if its scattering number is at most k. We also give an O(m+n) time algorithm for computing the scattering number of an interval graph with n vertices an m edges, which improves the O(n^4) time bound of Kratsch, Kloks and M\"uller. As a consequence of our two results the maximum k for which an interval graph is k-Hamilton-connected can be computed in O(m+n) time.