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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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This content will become publicly available on January 1, 2026
Probabilistic zero forcing with vertex reversion
Probabilistic zero forcing is a graph coloring process in which blue vertices infect (color blue) white vertices with a probability proportional to the number of neighboring blue vertices. This paper introduces reversion probabilistic zero forcing (RPZF), which shares the same infection dynamics but also allows for blue vertices to revert to being white in each round. A threshold number of blue vertices is produced such that the complete graph is entirely blue in the next round of RPZF with high probability. Utilizing Markov chain theory, a tool is formulated which, given a graph's RPZF Markov transition matrix, calculates the probability of whether the graph becomes all white or all blue as well as the time at which this is expected to occur.
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- Award ID(s):
- 1839918
- PAR ID:
- 10582125
- Publisher / Repository:
- International Linear Algebra Society
- Date Published:
- Journal Name:
- The Electronic Journal of Linear Algebra
- Volume:
- 41
- ISSN:
- 1081-3810
- Page Range / eLocation ID:
- 49 to 73
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Given a graph G, the zero forcing number of G, Z(G), is the minimum cardinality of any set S of vertices of 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. Hence the failed zero forcing number of a graph was defined to be the cardinality of the largest set of vertices which fails to force all vertices in the graph. A similar property called skew zero forcing was defined so that if there is exactly one neighbor u of v is not in S, then u is added to S in the next iteration. The difference is that vertices that are not in S can force other vertices. This leads to the failed skew zero forcing number of a graph, which is denoted by F−(G). In this paper, we provide a complete characterization of all graphs with F−(G)=1. Fetcie, Jacob, and Saavedra showed that the only graphs with a failed zero forcing number of 1 are either: the union of two isolated vertices; P3; K3; or K4. In this paper, we provide a surprising result: changing the forcing rule to a skew-forcing rule results in an infinite number of graphs with F−(G)=1.more » « less
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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.more » « less
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