arXiv 1204.62372012-04math.CO

图的概率迫零

Probabilistic Zero Forcing in Graphs

Cong X. Kang, Eunjeong Yi

备注：: 6 pages, 2 figures

摘要：

图$G$的\emph{zero forcing number} $Z(G)$是黑色顶点的集合$S$的最小基数（而$V(G)\setminusS$中的顶点被着色为白色），使得$V(G)$在多次应用“（经典）颜色变化规则”之后变为黑色：如果白色顶点是黑色顶点的唯一白色邻居，则白色顶点被转换为黑色顶点。“AIM最小秩-特殊图工作组”引入迫零数，并利用迫零数来界定图的最小图秩。我们在这里介绍了一个概率颜色变化规则（pccr），这是一个自然推广的经典颜色变化规则。我们引入了概率迫零产生的pccr的理论，该理论产生了一个数量$P_A(G)$，它可以被看作是一个图$G$与初始黑色集$A$将被完全转换为黑色的概率。我们还解释了这个理论的样本空间的演化作为一个马尔可夫过程。最后，我们用几个基本的例子来说明这个理论。

英文摘要：

The \emph{zero forcing number} $Z(G)$ of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)\setminusS$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the (classical) color change rule": a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. Zero forcing number was introduced and used to bound the minimum rank of graphs by the "AIM Minimum Rank - Special Graphs Work Group". We introduce here a probabilistic color change rule (pccr) which is a natural generalization of the classical color change rule. We introduce a theory of probabilistic zero forcing arising out of the pccr; the theory yields a quantity $P_A(G)$, which can be viewed as the probability that a graph $G$ with an initial black set $A$ will be converted entirely to the color black. We also interpret the evolution of the sample spaces of this theory as a Markov process. We end with a few basic examples illustrating this theory.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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