Lower bounds for positive semidefinite zero forcing and their applications

Yang, Boting

doi:10.1007/s10878-015-9936-0

Cited by 6 publications

(1 citation statement)

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“…Other generalizations and extensions of power domination and zero forcing have also been explored; for example, variants of the problems obtained by modifying the color change rules have received significant attention [8,9,14,26], as has the problem of studying the number of timesteps involved in the process of zero forcing or power dominating a graph [13,20,21].…”

Section: While There Existsmentioning

confidence: 99%

Restricted power domination and zero forcing problems

et al. 2018

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Power domination in graphs arises from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A power dominating set of a graph is a set of vertices that observes every vertex in the graph, following a set of rules for power system monitoring. A practical problem of interest is to determine the minimum number of additional measurement devices needed to monitor a power network when the network is expanded and the existing devices remain in place. In this paper, we study the problem of finding the smallest power dominating set that contains a given set of vertices X. We also study the related problem of finding the smallest zero forcing set that contains a given set of vertices X. The sizes of such sets in a graph G are respectively called the restricted power domination number and restricted zero forcing number of G subject to X. We derive several tight bounds on the restricted power domination and zero forcing numbers of graphs, and relate them to other graph parameters. We also present exact and algorithmic results for computing the restricted power domination number, including integer programs for general graphs and a linear time algorithm for graphs with bounded treewidth. We also use restricted power domination to obtain a parallel algorithm for finding minimum power dominating sets in trees.

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Section: While There Existsmentioning

confidence: 99%

Restricted power domination and zero forcing problems

et al. 2018

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An integer program for positive semidefinite zero forcing in graphs

2020

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Positive semidefinite (PSD) zero forcing is a dynamic graph process in which an initial subset of vertices are colored and may cause additional vertices to become colored through a set of color changing rules. Subsets which cause all other vertices to become colored are called PSD zero forcing sets; the PSD zero forcing number of a graph is the minimum cardinality attained by its PSD zero forcing sets. The PSD zero forcing number is of particular interest as it bounds solutions for the minimum rank and PSD min rank problems, both popular in linear algebra. This paper introduces blocking sets for PSD zero forcing sets which are used to formulate the first integer program (IP) for computing PSD zero forcing numbers of general graphs. It is shown that facets of the feasible region of this IP's linear relaxation correspond to zero forcing forts which induce connected subgraphs, but that identifying min cardinality connected forts is NP‐hard in general. Auxiliary IPs used to find these blocking sets are also given, enabling the master IP to be solved via constraint generation. Experiments comparing the proposed methods and existing algorithms are provided demonstrating improved runtime performance, particularly so in dense and sparse graphs.

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