All Graphs with a Failed Zero Forcing Number of Two

Gómez, Luis A.; Rubi, Karla; Terrazas, Jorden; Narayan, Darren A.

doi:10.3390/sym13112221

Cited by 6 publications

(7 citation statements)

References 8 publications

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“…We can sum the rows to get |F 2 | = 12, |F 3 | = 62, and |F 4 | = 641. This confirms the result by [3] where it was shown that there are 15 graphs with a failed zero forcing number of two, twelve of which were connected.…”

Section: Discussionsupporting

confidence: 90%

“…In this case, we can add any number of edges to G, and S will still be a failed zero forcing set. This is a special case of lemma 3 in [3].…”

Section: Graphs With Minimum Degree At Least Threementioning

confidence: 86%

“…The bound makes it possible in principle to classify all graphs with a given failed zero forcing number k by exhaustively checking all graphs such that n ≤ 2k + 2. This classification was done in [2] for k = 0, 1, and in [3] for k = 2, the latter using a special case of the bound proven here.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Ufferman¹,

Swanson²

2022

Preprint

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Given a graph G = (V, E) and a set of vertices marked as filled, we consider a color-change rule known as zero forcing. A set S is a zero forcing set if filling S and applying all possible instances of the color change rule causes all vertices in V to be filled. A failed zero forcing set is a set of vertices that is not a zero forcing set. Given a graph G, the failed zero forcing number F (G) is the maximum size of a failed zero forcing set. In [2], the authors asked whether given any k there is a an ℓ such that all graphs with at least ℓ vertices must satisfy F (G) ≥ k. We answer this question affirmatively by proving that for a graph G with n vertices, F (G) ≥ ⌊ n−1 2 ⌋.

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Section: Discussionsupporting

confidence: 90%

“…In this case, we can add any number of edges to G, and S will still be a failed zero forcing set. This is a special case of lemma 3 in [3].…”

Section: Graphs With Minimum Degree At Least Threementioning

confidence: 86%

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

Ufferman¹,

Swanson²

2022

Preprint

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“…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%

“…In 2021, Gomez, Rubi, Terrazas, and Narayan [7] identified the 15 graphs with a failed zero forcing number of two. These are shown in Figure 1.…”

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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