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

Abstract

Abstract.We show 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(n + m) time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the O(n 3 ) time bound of Kratsch, Kloks and Müller. As a consequence of our two results the maximum k for which an interval graph is kHamilton-connected can be computed in O(n + m) time.

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

“…By the latter statement, the corresponding optimization problem Spanning Connectivity is ‐hard. We posed as an open problem to determine the complexity of this problem for proper interval graphs and interval graphs . In response, very recently, Li and Wu announced an time algorithm for solving Spanning Connectivity on interval graphs.Kratsch, Kloks, and Müller gave an time algorithm for solving Toughness on interval graphs.…”

“…Contract grant sponsor: Royal Society Joint Project Grant JP090172. An extended abstract of it appeared in the Proceedings of WG 2013 [];…”

Comparing the speed and accuracy of approaches to betweenness centrality approximation

“…Computational aspects of many important node-based resilience measures are considered in [60], including vertex attack tolerance (VAT) [61,62], integrity [55], tenacity [63,64], toughness [65], and scattering number [66]. As all of these measures have associated computational hardness results, the performance of heuristics was considered on several representative networks in [60].…”

Computing the Weighted Isolated Scattering Number of Interval Graphs in Polynomial Time

“…Clearly, this weighted variant reduces to the original isolated scattering number when all vertex weights are equal to 1. While computing the scattering number is NP-hard in general [5], the scattering number can be computed in linear time for interval graphs [6].…”