Abstract A dynamic coloring of a graph G starts with an initial subset F ⊆ V ( G ) of colored vertices, while all the remaining vertices are non-colored. At each time step, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set F is called a zero forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. The zero forcing number of G , denoted by F ( G ) , is the cardinality of a minimum zero forcing set of G . The maximum nullity of G , denoted by M ( G ) , is the largest possible nullity over all | V ( G ) | by | V ( G ) | real symmetric matrices A whose non-diagonal entries are non-zero if the corresponding vertices are adjacent in G and with no restriction for its diagonal entries. In this paper, we characterize all graphs G of order n , maximum degree at most three, and F ( G ) = 3 . Also we classify these graphs with their maximum nullity.

中文翻译：

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最大度数最多为 3 的图上的最大无效和迫零数

摘要 图 G 的动态着色从着色顶点的初始子集 F ⊆ V ( G ) 开始，而所有剩余的顶点都是非着色的。在每个时间步长，一个有颜色的顶点正好有一个无色的邻居会强制这个无色的邻居被着色。如果通过迭代应用强制过程，G 中的每个顶点都被着色，则初始集 F 称为 G 的迫零集。G 的迫零数，用 F(G) 表示，是 G 的最小迫零集的基数。G 的最大无效性，用 M ( G ) 表示，是所有可能的最大无效性 | V ( G ) | 通过 | V ( G ) | 如果对应的顶点在 G 中相邻并且对其对角线项没有限制，则实对称矩阵 A 的非对角线项为非零。在本文中，我们刻画了所有 n 阶图 G，最大度数最多为 3，并且 F ( G ) = 3 。我们还用它们的最大无效性对这些图进行分类。