图的连通强迫数的刻画
Characterizations of the Connected Forcing Number of a Graph
摘要：
迫零是一个动态的图着色过程，其中具有单个未着色邻居的着色顶点强制该邻居着色。这种强迫过程已被用来近似某些线性代数参数，以及疾病和信息在社交网络中的传播模型。本文引入并研究了初着色顶点集诱导连通子图的迫零过程--连通迫零过程的一个限制。我们表明，连接强迫数-的最小的初始着色顶点集，迫使整个图着色的基数-是一个尖锐的上限的最大零，路径覆盖数，和叶子数的图。我们还给出了封闭公式和几个家庭的图，包括树，超立方体，花蛇的连接强迫数的界限，并与极值连接强迫数的特征图。
英文摘要：
Zero forcing is a dynamic graph coloring process whereby a colored vertex with a single uncolored neighbor forces that neighbor to be colored. This forcing process has been used to approximate certain linear algebraic parameters, as well as to model the spread of diseases and information in social networks. In this paper, we introduce and study the connected forcing process -- a restriction of zero forcing in which the initially colored set of vertices induces a connected subgraph. We show that the connected forcing number -- the cardinality of the smallest initially colored vertex set which forces the entire graph to be colored -- is a sharp upper bound to the maximum nullity, path cover number, and leaf number of the graph. We also give closed formulas and bounds for the connected forcing numbers of several families of graphs including trees, hypercubes, and flower snarks, and characterize graphs with extremal connected forcing numbers.
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