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

Barioli, Francesco; Barrett, Wayne; Fallat, Shaun M.; Hall, H. Tracy; Hogben, Leslie; Shader, Bryan L.; Driessche, P. van den; Holst, Hein van der

doi:10.1002/jgt.21637

Cited by 111 publications

(161 citation statements)

References 32 publications

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“…There are two ways to make P an (α, β)linkage (up to swapping α and β). For α = {1, 3} and β = {2, 4}, P is not (α, β)-rigid because P ′ = { (1,4), (3,2)} is another (α, β)-linkage. Forα = {1, 4} andβ = {2, 3}, P is not (α,β)-rigid because P ′ = {(1, 3), (4, 2)} is another (α,β)-linkage.…”

Section: Introductionmentioning

confidence: 99%

Rigid Linkages and Partial Zero Forcing

Ferrero¹,

Flagg²,

Hall³

et al. 2019

Electron. J. Combin.

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Connections between vital linkages and zero forcing are established. Specifically, the notion of a rigid linkage is introduced as a special kind of unique linkage and it is shown that spanning forcing paths of a zero forcing process form a spanning rigid linkage and thus a vital linkage. A related generalization of zero forcing that produces a rigid linkage via a coloring process is developed. One of the motivations for introducing zero forcing is to provide an upper bound on the maximum multiplicity of an eigenvalue among the real symmetric matrices described by a graph.1 Rigid linkages and a related notion of rigid shortest linkages are utilized to obtain bounds on the multiplicities of eigenvalues of this family of matrices.

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Section: Introductionmentioning

confidence: 99%

Rigid Linkages and Partial Zero Forcing

Ferrero¹,

Flagg²,

Hall³

et al. 2019

Electron. J. Combin.

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“…It is also interesting to point out some connections between our work and other well‐known graph‐theoretic problems. In particular, the so‐called zero‐forcing problem (and the strictly related concept of power domination ) are dynamical processes in graphs whose evolution can be described within our framework. More precisely, it can be easily seen that they correspond to an evangelization process (to use our language) in networks whose thresholds

t_{I} (v)

are all equal to one.…”

Section: What Is Already Known and What We Provementioning

confidence: 99%

Evangelism in social networks: Algorithms and complexity

et al. 2017

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We consider a population of interconnected individuals that, with respect to a piece of information, at each time instant can be subdivided into three (time-dependent) categories: agnostics, influenced, and evangelists. A dynamical process of information diffusion evolves among the individuals of the population according to the following rules. Initially, all individuals are agnostic. Then, a set of people is chosen from the outside and convinced to start evangelizing, i.e., to start spreading the information. When a number of evangelists, greater than a given threshold, communicate with a node v, the node v becomes influenced, whereas, as soon as the individual v is contacted by a sufficiently much larger number of evangelists, it is itself converted into an evangelist and consequently it starts spreading the information. The question is: How to choose a bounded cardinality initial set of evangelists so as to maximize the final number of influenced individuals? We prove that the problem is hard to solve, even in an approximate sense. On the positive side, we present exact polynomial time algorithms for trees and complete graphs. For general graphs, we derive exact parameterized algorithms. We also investigate the problem when the objective is to select a minimum number of evangelists capable of influencing the whole network. Our motivations to study these problems come from the areas of Viral Marketing and the analysis of quantitative models of spreading of influence in social networks.

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“…For a graph G, it is well known that if K r is a subgraph of G then M(G) ≥ r − 1 (see, for example, [Barioli et al 2013] and the references therein). An analogous result holds true for digraphs, although the proof is different than those usually given for graphs.…”

Section: Digraphs In Generalmentioning

confidence: 99%

Path cover number, maximum nullity, and zero forcing number of oriented graphs and other simple digraphs

Berliner

Brown

Carlson

et al. 2015

Involve

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An oriented graph is a simple digraph obtained from a simple graph by choosing exactly one of the two arcs (u, v) or (v, u) to replace each edge {u, v}. A simple digraph describes the zero-nonzero pattern of off-diagonal entries of a family of (not necessarily symmetric) matrices. The minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number and path cover number are related parameters. We establish bounds on the range of possible values of all these parameters for oriented graphs, establish connections between the values of these parameters for a simple graph G, for various orientations G and for the doubly directed digraph of G, and establish an upper bound on the number of arcs in a simple digraph in terms of the zero forcing number.

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

Cited by 111 publications

References 32 publications

Rigid Linkages and Partial Zero Forcing

Rigid Linkages and Partial Zero Forcing

Evangelism in social networks: Algorithms and complexity

Path cover number, maximum nullity, and zero forcing number of oriented graphs and other simple digraphs

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