论文标题
偏斜零强迫数的表征1
Characterization of Graphs With Failed Skew Zero Forcing Number of 1
论文作者
论文摘要
给定图表$ g $，零强迫数为$ g $，$ z（g）$，是任何集合$ s $ s $顶点的基数最小的基数，在上面，强迫规则的重复应用在所有顶点中都在$ s $中导致所有顶点。强迫规则是：如果一个顶点$ v $在$ s $中，而恰好一个邻居$ u $ o $ v $不在$ s $中，则$ u $在下一个迭代中添加到$ s $中。因此，将图的零强迫数定义为最大的顶点集的大小，该顶点无法强制图中的所有顶点。定义了一个类似的称为偏斜强迫的属性，因此，如果完全存在一个邻居$ u $ o $ v $不在$ s $中，则$ u $将在下一个迭代中添加到$ s $中。不同之处在于，不在$ s $中的顶点会迫使其他顶点。这导致了失败的偏斜零强迫数量，该图由$ f^{ - }（g）$表示。在本文中，我们提供了所有图表的完整表征，其中$ f^{ - }（g）= 1 $。 Fetcie，Jacob和Saavedra表明，唯一的零强迫数量为$ 1 $的图是：两个孤立顶点的结合； $ P_3 $; $ k_3 $;或$ k_4 $。在本文中，我们提供了一个令人惊讶的结果：将强制规则更改为偏斜规则导致无限数量的图形数量，其中$ f^{ - }（g）= 1 $。
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. Hence the failed zero forcing number of a graph was defined to be the size 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; $P_3$; $K_3$; or $K_4$. 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 $F^{-}(G)=1$.
