Positive semidefinite zero forcing

Ekstrand, Jason; Erickson, Craig; Hall, H. Tracy; Hay, Diana; Hogben, Leslie; Johnson, Ryan; Kingsley, Nicole; Osborne, Steven; Peters, Travis; Roat, Jolie; Ross, Arianne; Row, Darren D.; Warnberg, Nathan; Young, Michael E.

doi:10.1016/j.laa.2013.05.020

Cited by 43 publications

(45 citation statements)

References 16 publications

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“…A minimum tree cover of G is a tree cover of G whose cardinality is equal to tc(G). Barioli et al (2011) showed that for any outerplanar graph G, M + (G) = tc(G); and Ekstrand et al (2013) showed that for any graph G, tc(G) ≤ Z + (G).…”

Section: Preliminariesmentioning

confidence: 99%

“…The most widely-studied variant of zero forcing, called the positive semidefinite zero forcing (PSZF for short), was introduced in Barioli et al (2010) [see also Ekstrand et al (2012Ekstrand et al ( , 2013; Fallat et al (2015) for recent results]. Similar to the zero forcing number, a primary reason to study the PSZF number of a graph is its relationship to the maximum positive semidefinite nullity of certain positive semidefinite symmetric matrices associated with the graph.…”

Section: Introductionmentioning

confidence: 99%

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Lower bounds for positive semidefinite zero forcing and their applications

Yang

2015

J Comb Optim

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The positive semidefinite zero forcing number of a graph is a parameter that is important in the study of minimum rank problems. In this paper, we focus on the algorithmic aspects of computing this parameter. We prove that it is NP-complete to find the positive semidefinite zero forcing number of a given graph, and this problem remains NP-complete even for graphs with maximum vertex degree 7. We present a linear time algorithm for computing the positive semidefinite zero forcing number of generalized series-parallel graphs. We introduce the constrained tree cover number and apply it to improve lower bounds for positive semidefinite zero forcing. We also give formulas for the constrained tree cover number and the tree cover number on graphs with special structures.Keywords Zero forcing number · Minimum rank · Tree cover number · Positive semidefinite zero forcing number · Maximum positive semidefinite nullity

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lower bounds for positive semidefinite zero forcing and their applications

Yang

2015

J Comb Optim

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“…That the upper bounds are correct is immediate. The result n − 2 ≤ Z(G) + Z(G) appears in [7]. Then n − 3 ≤ Z(G) · Z(G) follows, because 1 ≤ Z(G) for all G and the function f (z) = z(n − 2 − z) attains its minimum on the interval [1, n − 3] at the endpoints.…”

Section: Introductionmentioning

confidence: 96%

Nordhaus–Gaddum problems for power domination

Benson

Ferrero

Flagg

et al. 2018

Discrete Applied Mathematics

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The upper and lower Nordhaus-Gaddum bounds over all graphs for the power domination number follow from known bounds on the domination number and examples. In this note we improve the upper sum bound for the power domination number substantially for graphs having the property that both the graph and its complement must be connected. For these graphs, our bound is tight and is also significantly better than the corresponding bound for domination number. We also improve the product upper bound for the power domination number for graphs with certain properties.

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“…Namely, the positive semidefinite zero forcing number of G, denoted Z + (G), is defined for any graph G. While a precise definition of Z + (G) is beyond the scope of this paper, we note that the definition is purely combinatorial, so that Z + (G) may be considered from a strictly graph-theoretic perspective. (See, e.g., [11].) Nevertheless, msr(G) ≥ n − Z + (G) for every graph G, though the gap may be arbitrarily large [18].…”

Section: Introductionmentioning

confidence: 99%

Orthogonal representations of Steiner triple system incidence graphs

Deaett

Hall

2017

Discrete Mathematics

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Abstract. The unique Steiner triple system of order 7 has a point-block incidence graph known as the Heawood graph. Motivated by questions in combinatorial matrix theory, we consider the problem of constructing a faithful orthogonal representation of this graph, i.e., an assignment of a vector in C d to each vertex such that two vertices are adjacent precisely when assigned nonorthogonal vectors. We show that d = 10 is the smallest number of dimensions in which such a representation exists, a value known as the minimum semidefinite rank of the graph, and give such a representation in 10 real dimensions. We then show how the same approach gives a lower bound on this parameter for the incidence graph of any Steiner triple system, and highlight some questions concerning the general upper bound.

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Positive semidefinite zero forcing

Cited by 43 publications

References 16 publications

Lower bounds for positive semidefinite zero forcing and their applications

Lower bounds for positive semidefinite zero forcing and their applications

Nordhaus–Gaddum problems for power domination

Orthogonal representations of Steiner triple system incidence graphs

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