Parameters Related to Tree-Width, Zero Forcing, and Maximum Nullity of a Graph

Date
2013-02-01
Authors
Barioli, Francesco
Barrett, Wayne
Fallat, Shaun
Hogben, Leslie
Hogben, Leslie
Shader, Bryan
van den Driessche, P.
van der Holst, Hein
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Mathematics
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Abstract

Tree-width, and variants that restrict the allowable tree decompositions, play an important role in the study of graph algorithms and have application to computer science. The zero forcing number is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by a graph. We establish relationships between these parameters, including several Colin de Verdière type parameters, and introduce numerous variations, including the minor monotone floors and ceilings of some of these parameters. This leads to new graph parameters and to new characterizations of existing graph parameters. In particular, tree-width, largeur d'arborescence, path-width, and proper path-width are each characterized in terms of a minor monotone floor of a certain zero forcing parameter defined by a color change rule.

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<p>This is the peer-reviewed version of the following article: Barioli, Francesco, Wayne Barrett, Shaun M. Fallat, H. Tracy Hall, Leslie Hogben, Bryan Shader, Pauline van den Driessche, and Hein Van Der Holst. "Parameters Related to Tree‐Width, Zero Forcing, and Maximum Nullity of a Graph." <em>Journal of Graph Theory</em> 72, no. 2 (2013): 146-177, which has been published in final form at DOI: <a href="https://doi.org/10.1002/jgt.21637" target="_blank">10.1002/jgt.21637</a>. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Posted with permission.</p>
Keywords
tree-width, path-width, zero forcing number, maximum nullity, minimum rank, Colin de Verdiere type parameter, minor monotone floor, minor monotone ceiling
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