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


  • This research began at the American Institute of Mathematics SQuaRE, “Minimum Rank of Symmetric Matrices Described by a Graph,” and the authors thank AIM and NSF for their support.Research of SMF and PvdD supported in part by NSERC Discovery grants.


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.