On the zero blocking number of rectangular, cylindrical, and Möbius grids

Beaudouin-Lafon, Matthew; Crawford, Margaret; Chen, Serena; Karst, Nathaniel J.; Nielsen, L. D.; Troxell, Denise Sakai

doi:10.1016/j.dam.2019.11.015

Cited by 6 publications

(5 citation statements)

References 21 publications

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“…Shitov [4], proved that determining the failed zero forcing number of a graph is NP-complete. Independently, a closely related property called the zero blocking number of a graph was introduced in 2020 by Beaudouin-Lafona, Crawford, Chen, Karst, Nielsen, and Sakai Troxell [5] and Karst, Shen, and Vu [6]. The zero blocking number of a graph G equals |V(G)| − F(G).…”

Section: Introductionmentioning

confidence: 99%

All Graphs with a Failed Zero Forcing Number of Two

et al. 2021

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Given a graph G, the zero forcing number of G, Z(G), is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being in S. The forcing rule is: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. Zero forcing numbers have attracted great interest over the past 15 years and have been well studied. In this paper, we investigate the largest size of a set S that does not force all of the vertices in a graph to be in S. This quantity is known as the failed zero forcing number of a graph and will be denoted by F(G). We present new results involving this parameter. In particular, we completely characterize all graphs G where F(G)=2, solving a problem posed in 2015 by Fetcie, Jacob, and Saavedra.

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Section: Introductionmentioning

confidence: 99%

All Graphs with a Failed Zero Forcing Number of Two

et al. 2021

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“…Recently, Ufferman and Swanson [4] presented lower and upper bounds for F(G). Independently, the zero blocking number of a graph which equals |V(G)| − F(G) was introduced in 2020 by Beaudouin-Lafona, Crawford, Chen, Karst, Nielsen, and Sakai Troxell [5] and Karst, Shen, and Vu [6].…”

Section: Introductionmentioning

confidence: 99%

“…The first problem has been well studied with results appearing in [2,[4][5][6][7]. The second "inverse" type problem has only been investigated where the failed zero forcing set is a single vertex.…”

mentioning

confidence: 99%

An Inverse Approach for Finding Graphs with a Failed Zero Forcing Number of k

Kaudan,

Taylor,

Narayan

2023

Mathematics

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For a given a graph G, the zero forcing number of G, Z(G), is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being included in S. The forcing rule is as follows: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. The failed zero forcing number of a graph was introduced by Fetcie, Jacob, and Saavedra and was defined as the cardinality of the largest set of vertices which fails to force all vertices in the graph. In 2021, Gomez et al. proved that there were exactly 15 graphs with a failed zero forcing number of two, but their proof was complicated, requiring the analysis of many graph families. We present an “inverse” approach where we start with a set of vertices S and then see which graphs have S as a maximum-sized failed zero forcing set. This results in not only a shorter proof but characterizes which graphs have a particular failed zero forcing set. In our proof, we characterize the graphs which have a failed zero forcing set S where |S|=2, meaning considering all simple graphs of order two as induced subgraphs. This approach also has greater potential for characterizing graphs where F(G)>2 since many general arguments on the structure of graphs are developed in this type of analysis and are applicable for failed zero forcing sets of any size.

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“…In this paper we investigate the largest size of a set S that does not force all of the vertices in a graph to be in S. This quantity is known as the failed zero forcing number of a graphs and will be denoted by F (G), and has received attention in recent years [3]. Independently, a closely related property called the zero blocking number of a graph was introduced by Beaudouin-Lafona, Crawford, Chen, Karst, Nielsen, and Sakai Troxell [5] and Karst, Shen, and Vu [4]. The zero blocking number of a graph G equals |V (G)| − F (G).…”

Section: Introductionmentioning

confidence: 99%

All Graphs with a Failed Zero Forcing Number of Two

Gómez¹,

Rubi²,

Terrazas³

et al. 2021

Preprint

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Given a graph G, the zero-forcing number of G, Z(G), is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being in S. The forcing rule is: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. Zero-forcing numbers have attracted great interest over the past 15 years and have been well studied. In this paper we investigate the largest size of a set S that does not force all of the vertices in a graph to be in S. This quantity is known as the failed zero-forcing number of a graphs and will be denoted by F (G), and has received attention in recent years. We present new results involving this parameter. In particular, we completely characterize all graphs G where F (G) = 2, solving a problem posed in 2015 by Fetcie, Jacob, and Saavedra.

show abstract

On the zero blocking number of rectangular, cylindrical, and Möbius grids

Cited by 6 publications

References 21 publications

All Graphs with a Failed Zero Forcing Number of Two

All Graphs with a Failed Zero Forcing Number of Two

An Inverse Approach for Finding Graphs with a Failed Zero Forcing Number of k

All Graphs with a Failed Zero Forcing Number of Two

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