Linear‐Time Algorithms for Scattering Number and Hamilton‐Connectivity of Interval Graphs1. Issue 4 (28th October 2014)

• Record Type: Journal Article
• Authors: Broersma, Hajo; Fiala, Jiří; Golovach, Petr A.; Kaiser, Tomáš; Paulusma, Daniël; Proskurowski, Andrzej
• Is Part Of: Journal of graph theory. Volume 79:Issue 4(2015)
• Page Start: 282
• Page End: 299
• Publication Date: 2014-10-28
• Subjects: Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.21832
• Languages: English
• ISSNs: 0364-9024

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We prove that for all <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjrqj4w67" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21832:jgt21832-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>≤</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> an interval graph is <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjrqj4w7r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21832:jgt21832-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>−</mml:mo><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>‐Hamilton‐connected if and only if its scattering number is at most <italic>k</italic>. 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 <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjrqj4w88" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21832:jgt21832-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> time algorithm for computing the scattering number of an interval graph with <italic>n</italic> vertices and <italic>m</italic> edges, which improves the previously best‐known <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjrqj4w1n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21832:jgt21832-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> time bound for solving this problem. As a consequence of our two results, the maximum <italic>k</italic> for which an interval graph is <italic>k</italic>‐Hamilton‐connected can be computed in <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjrqj4w25" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21832:jgt21832-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> time.</p> </abstract>