A New Lower Bound for Positive Zero Forcing

2017 ◽  
pp. 254-266
Author(s):  
Boting Yang
Keyword(s):  
Lower Bound ◽  
Positive Zero ◽  
Zero Forcing

scholarly journals A lower bound on the zero forcing number

2018 ◽  
Vol 250 ◽  
pp. 363-367 ◽  
Author(s):  
Randy Davila ◽  
Thomas Kalinowski ◽  
Sudeep Stephen
Keyword(s):  
Lower Bound ◽  
Zero Forcing ◽  
Zero Forcing Number ◽  
Forcing Number

scholarly journals Techniques for determining equality of the maximum nullity and the zero forcing number of a graph

2021 ◽  
Vol 37 ◽  
pp. 295-315
Author(s):  
Derek Young
Keyword(s):  
Lower Bound ◽  
Adjacency Matrix ◽  
Work Group ◽  
Zero Forcing ◽  
Vertex Connectivity ◽  
Aztec Diamond ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
Forcing Number ◽  
Graph Parameters

It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph (see [AIM Minimum Rank - Special Graphs Work Group (F. Barioli, W. Barrett, S. Butler, S. Cioab$\breve{\text{a}}$, D. Cvetkovi$\acute{\text{c}}$, S. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi$\acute{\text{c}}$, H. van der Holst, K. Vander Meulen, and A. Wangsness). Linear Algebra Appl., 428(7):1628--1648, 2008]). In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number of the graph are the same by studying a variety of graph parameters that give lower bounds on the maximum nullity of a graph. Inparticular, we introduce a new graph parameter which acts as a lower bound for the maximum nullity of the graph. As a result, we show that the Aztec Diamond graph's maximum nullity and zero forcing number are the same. Other graph parameters that are considered are a Colin de Verdiére type parameter and vertex connectivity. We also use matrices, such as a divisor matrix of a graph and an equitable partition of the adjacency matrix of a graph, to establish a lower bound for the nullity of the graph's adjacency matrix.

Positive Zero Forcing and Edge Clique Coverings

2016 ◽  
pp. 53-64 ◽  
Author(s):  
Shaun Fallat ◽  
Karen Meagher ◽  
Abolghasem Soltani ◽  
Boting Yang
Keyword(s):  
Positive Zero ◽  
Zero Forcing

scholarly journals On the Relationships between Zero Forcing Numbers and Certain Graph Coverings

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Fatemeh Alinaghipour Taklimi ◽  
Shaun Fallat ◽  
Karen Meagher
Keyword(s):  
Upper Bound ◽  
Tree Cover ◽  
Positive Zero ◽  
Zero Forcing ◽  
Outerplanar Graphs ◽  
Zero Forcing Number ◽  
Forcing Number ◽  
Minimum Number ◽  
Graph Parameters ◽  
Graph Coverings

AbstractThe zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph that cover all the vertices of the graph, while the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph needed to cover all the vertices in the graph. We show that for a block-cycle graph the zero forcing number equals the path cover number.We also give a purely graph theoretical proof that the positive zero forcing number of any outerplanar graphs equals the tree cover number of the graph. These ideas are then extended to the setting of k-trees, where the relationship between the positive zero forcing number and the tree cover number becomes more complex.

scholarly journals Compressed cliques graphs, clique coverings and positive zero forcing

2018 ◽  
Vol 734 ◽  
pp. 119-130 ◽  
Author(s):  
Shaun Fallat ◽  
Karen Meagher ◽  
Abolghasem Soltani ◽  
Boting Yang
Keyword(s):  
Positive Zero ◽  
Zero Forcing

scholarly journals A short proof for a lower bound on the zero forcing number

2020 ◽  
Vol 40 (1) ◽  
pp. 355 ◽  
Author(s):  
Maximilian Fürst ◽  
Dieter Rautenbach
Keyword(s):  
Lower Bound ◽  
Short Proof ◽  
Zero Forcing ◽  
Zero Forcing Number ◽  
Forcing Number

scholarly journals Edge Forcing in Butterfly Networks

2021 ◽  
Vol 182 (3) ◽  
pp. 285-299
Author(s):  
G. Jessy Sujana ◽  
T.M. Rajalaxmi ◽  
Indra Rajasingh ◽  
R. Sundara Rajan
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Higher Dimensions ◽  
Zero Forcing ◽  
Minimum Cardinality ◽  
Forcing Number ◽  
Np Complete ◽  
Forcing Set ◽  
Zero Forcing Set

A zero forcing set is a set S of vertices of a graph G, called forced vertices of G, which are able to force the entire graph by applying the following process iteratively: At any particular instance of time, if any forced vertex has a unique unforced neighbor, it forces that neighbor. In this paper, we introduce a variant of zero forcing set that induces independent edges and name it as edge-forcing set. The minimum cardinality of an edge-forcing set is called the edge-forcing number. We prove that the edge-forcing problem of determining the edge-forcing number is NP-complete. Further, we study the edge-forcing number of butterfly networks. We obtain a lower bound on the edge-forcing number of butterfly networks and prove that this bound is tight for butterfly networks of dimensions 2, 3, 4 and 5 and obtain an upper bound for the higher dimensions.

scholarly journals On the L-Grundy domination number of a graph

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3205-3215 ◽  
Author(s):  
Bostjan Bresar ◽  
Tanja Gologranc ◽  
Michael Henning ◽  
Tim Kos
Keyword(s):  
Lower Bound ◽  
Regular Graph ◽  
Decision Problem ◽  
Linear Time ◽  
Domination Number ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Zero Forcing ◽  
Split Graph ◽  
Np Complete

In this paper, we continue the study of the L-Grundy domination number of a graph introduced and first studied in [Grundy dominating sequences and zero forcing sets, Discrete Optim. 26 (2017) 66-77]. A vertex in a graph dominates itself and all vertices adjacent to it, while a vertex totally dominates another vertex if they are adjacent. A sequence of distinct vertices in a graph G is called an L-sequence if every vertex v in the sequence is such that v dominates at least one vertex that is not totally dominated by any vertex that precedes v in the sequence. The maximum length of such a sequence is called the L-Grundy domination number, L gr(G), of G. We show that the L-Grundy domination number of every forest G on n vertices equals n, and we provide a linear-time algorithm to find an L-sequence of length n in G. We prove that the decision problem to determine if the L-Grundy domination number of a split graph G is at least k for a given integer k is NP-complete. We establish a lower bound on L gr(G) when G is a regular graph, and investigate graphs G on n vertices for which L gr(G) = n.

Zero Forcing Beamforming Based Coordinated Scheduling Algorithm for Downlink Coordinated Multi-Point Transmission System

2015 ◽  
Vol E98.B (2) ◽  
pp. 352-359 ◽  
Author(s):  
Ping WANG ◽  
Lei DING ◽  
Huifang PANG ◽  
Fuqiang LIU ◽  
Nguyen Ngoc VAN
Keyword(s):  
Scheduling Algorithm ◽  
Transmission System ◽  
Zero Forcing ◽  
Coordinated Scheduling
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