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1. Characterization of All Graphs with a Failed Skew Zero Forcing Number of 1

   https://doi.org/10.3390/math10234463  

   Johnson, Aidan; Vick, Andrew E.; Narayan, Darren A. (December 2022, Mathematics)

   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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2. All Graphs with a Failed Zero Forcing Number of Two

   https://doi.org/10.3390/sym13112221  

   Gomez, Luis; Rubi, Karla; Terrazas, Jorden; Narayan, Darren A. (November 2021, Symmetry)

   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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3. Failed zero forcing and critical sets on directed graphs

   Adams, Alyssa; Jacob, Bonnie (October 2021, The Australasian journal of combinatorics)

   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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4. An integer program for positive semidefinite zero forcing in graphs

   https://doi.org/10.1002/net.21947  

   Smith, Logan A.; Mikesell, Derek J.; Hicks, Illya V. (May 2020, Networks)

   Abstract Positive semidefinite (PSD) zero forcing is a dynamic graph process in which an initial subset of vertices are colored and may cause additional vertices to become colored through a set of color changing rules. Subsets which cause all other vertices to become colored are called PSD zero forcing sets; the PSD zero forcing number of a graph is the minimum cardinality attained by its PSD zero forcing sets. The PSD zero forcing number is of particular interest as it bounds solutions for the minimum rank and PSD min rank problems, both popular in linear algebra. This paper introduces blocking sets for PSD zero forcing sets which are used to formulate the first integer program (IP) for computing PSD zero forcing numbers of general graphs. It is shown that facets of the feasible region of this IP's linear relaxation correspond to zero forcing forts which induce connected subgraphs, but that identifying min cardinality connected forts is‐hard in general. Auxiliary IPs used to find these blocking sets are also given, enabling the master IP to be solved via constraint generation. Experiments comparing the proposed methods and existing algorithms are provided demonstrating improved runtime performance, particularly so in dense and sparse graphs. 

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5. Quasi-polynomial time approximation schemes for the Maximum Weight Independent Set Problem in H-free graphs

   https://doi.org/10.1137/1.9781611975994.139  

   Chudnovsky, Maria.; Pillipczuk, Marcin.; Pillipczuk, Mihal; and Thomasse, Stephan. (April 2020, Proceedings of the annual ACMSIAM symposium on discrete algorithms)

   In the Maximum Independent Set problem we are asked to find a set of pairwise nonadjacent vertices in a given graph with the maximum possible cardinality. In general graphs, this classical problem is known to be NP-hard and hard to approximate within a factor of n1−ε for any ε > 0. Due to this, investigating the complexity of Maximum Independent Set in various graph classes in hope of finding better tractability results is an active research direction. In H-free graphs, that is, graphs not containing a fixed graph H as an induced subgraph, the problem is known to remain NP-hard and APX-hard whenever H contains a cycle, a vertex of degree at least four, or two vertices of degree at least three in one connected component. For the remaining cases, where every component of H is a path or a subdivided claw, the complexity of Maximum Independent Set remains widely open, with only a handful of polynomial-time solvability results for small graphs H such as P5, P6, the claw, or the fork. We prove that for every such “possibly tractable” graph H there exists an algorithm that, given an H-free graph G and an accuracy parameter ε > 0, finds an independent set in G of cardinality within a factor of (1 – ε) of the optimum in time exponential in a polynomial of log | V(G) | and ε−1. That is, we show that for every graph H for which Maximum Independent Set is not known to be APX-hard in H-free graphs, the problem admits a quasi-polynomial time approximation scheme in this graph class. Our algorithm works also in the more general weighted setting, where the input graph is supplied with a weight function on vertices and we are maximizing the total weight of an independent set. 

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