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Linear‐Time Algorithms for Scattering Number and Hamilton‐Connectivity of Interval Graphs
Author(s) -
Broersma Hajo,
Fiala Jiří,
Golovach Petr A.,
Kaiser Tomáš,
Paulusma Daniël,
Proskurowski Andrzej
Publication year - 2015
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.21832
Subject(s) - mathematics , combinatorics , interval (graph theory) , interval graph , hamiltonian path , discrete mathematics , graph , connectivity , path (computing) , pathwidth , line graph , computer science , programming language
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.