A Lower Bound on the Failed Zero Forcing Number of a Graph

Ufferman, Eric; Swanson, Nicolas

doi:10.48550/arxiv.2202.03821

Cited by 1 publication

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“…Shitov [3], proved that the problem of determining the failed zero forcing number of a given graph was NP-complete. 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%

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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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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.…”

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confidence: 99%