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

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\u27arborescence, 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

Tissue Polypeptide Antigen and Tissue Polypeptide Specific Antigen in Primary Breast Cancer. Evaluation in Serum and Tutmour Tissue

Peer Reviewe

On the Power Domination Number of de Bruijn and Kautz Digraphs

Let G=(V,A) be a directed graph, and let S⊆V be a set of vertices. Let the sequence S=S₀⊆S₁⊆S₂⊆⋯ be defined as follows: S₁ is obtained from S₀ by adding all out-neighbors of vertices in S₀. For k⩾2, Sₖ is obtained from Sₖ₋₁ by adding all vertices w such that for some vertex v∈Sₖ₋₁, w is the unique out-neighbor of v in V∖Sₖ₋₁. We set M(S)=S₀∪S₁∪⋯, and call S a power dominating set for G if M(S)=V(G). The minimum cardinality of such a set is called the power domination number of G. In this paper, we determine the power domination numbers of de Bruijn and Kautz digraphs

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

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.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." Journal of Graph Theory 72, no. 2 (2013): 146-177, which has been published in final form at DOI: 10.1002/jgt.21637. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Posted with permission.</p

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3

Zero forcing parameters and minimum rank problems

The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a graph G, is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by G. It is shown that for a connected graph of order at least two, no vertex is in every zero forcing set. The positive semidefinite zero forcing number Z+(G) is introduced, and shown to be equal to |G|-OS(G), where OS(G) is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank. The positive semidefinite zero forcing number is applied to the computation of positive semidefinite minimum rank of certain graphs. An example of a graph for which the real positive symmetric semidefinite minimum rank is greater than the complex Hermitian positive semidefinite minimum rank is presented.This is a manuscript of an article from Linear Algebra and its Applications 433 (2010): 401, doi:10.1016/j.laa.2010.03.008. Posted with permission.</p