论文标题
在平面图和无爪立方图中，零零强迫和grundy统治
Loop zero forcing and grundy domination in planar graphs and claw-free cubic graphs
论文作者
论文摘要
给定一个带有顶点套装$ V（g）$的简单有限的图形，我们定义了零g $的零强迫集。选择$ s \ subseteq v（g）$，并在$ v（g） - s $白色中颜色$ s $ blue的所有顶点和所有顶点。颜色更改规则是，如果$ w $是蓝色顶点$ v $的唯一白色邻居，那么我们将$ w $的颜色从白色更改为蓝色。如果在应用颜色更改规则后尽可能多次，最终每个顶点$ g $都是蓝色的，我们将$ s $称为零g $的零强迫集。 $ z（g）$表示强迫集的最小基数。 Davila和Henning在\ cite {zerocubic}中证明了任何无爪的立方图$ g $，$ z（g）\ le \ frac {1} {3} {3} | v（g）| + 1 $。我们表明，如果$ g $是$ 2 $ - 边缘连接，无爪和立方，则$ z（g）\ le \ left \ lceil \ frac \ frac {5n（g）} {18} {18} \ right \ right \ rceil+1 $。我们还研究了一个类似的图形不变式，称为循环零强迫$ g $的强迫数量，这恰好是$ g $的Grundy Donination数字的双重不变。具体而言，我们研究了两种特定类型的平面图中的零零强迫数量。
Given a simple, finite graph with vertex set $V(G)$, we define a zero forcing set of $G$ as follows. Choose $S\subseteq V(G)$ and color all vertices of $S$ blue and all vertices in $V(G) - S$ white. The color change rule is if $w$ is the only white neighbor of blue vertex $v$, then we change the color of $w$ from white to blue. If after applying the color change rule as many times as possible eventually every vertex of $G$ is blue, we call $S$ a zero forcing set of $G$. $Z(G)$ denotes the minimum cardinality of a zero forcing set. Davila and Henning proved in \cite{zerocubic} that for any claw-free cubic graph $G$, $Z(G) \le \frac{1}{3}|V(G)| + 1$. We show that if $G$ is $2$-edge-connected, claw-free, and cubic, then $Z(G) \le \left\lceil\frac{5n(G)}{18}\right\rceil+1$. We also study a similar graph invariant known as the loop zero forcing number of a graph $G$ which happens to be the dual invariant to the Grundy domination number of $G$. Specifically, we study the loop zero forcing number in two particular types of planar graphs.
