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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.
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Failed zero forcing and critical sets on directed graphs
Let $$D$$ be a simple digraph (directed graph) with vertex set $V(D)$ and arc set $A(D)$ where $n=|V(D)|$, and each arc is an ordered pair of distinct vertices. If $$(v,u) \in A(D)$$, then $$u$$ is considered an \emph{out-neighbor} of $$v$$ in $$D$$. Initially, we designate each vertex to be either filled or empty. Then, the following color change rule (CCR) is applied: if a filled vertex $$v$$ has exactly one empty out-neighbor $$u$$, then $$u$$ will be filled. If all vertices in $V(D)$$ are eventually filled under repeated applications of the CCR, then the initial set is called a \emph{zero forcing set} (ZFS); if not, it is a \emph{failed zero forcing set} (FZFS). We introduce the \emph{failed zero forcing number} $$\F(D)$$ on a digraph, which is the maximum cardinality of any FZFS. The \emph{zero forcing number}, $$\Z(D)$, is the minimum cardinality of any ZFS. We characterize digraphs that have $$\F(D)<\Z(D)$$ and determine $$\F(D)$$ for several classes of digraphs including de Bruijn and Kautz digraphs. We also characterize digraphs with $$\F(D)=n-1$$, $$\F(D)=n-2$$, and $$\F(D)=0$$, which leads to a characterization of digraphs in which any vertex is a ZFS. Finally, we show that for any integer $$n \geq 3$$ and any non-negative integer $$k$$ with $k
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- Award ID(s):
- 1659299
- PAR ID:
- 10354464
- Date Published:
- Journal Name:
- The Australasian journal of combinatorics
- Volume:
- 81
- Issue:
- 3
- ISSN:
- 2202-3518
- Page Range / eLocation ID:
- 367-387
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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.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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