图的动态垄断：平均和严格多数阈值
On dynamic monopolies of graphs: the average and strict majority thresholds
摘要：
令$G$是一个图，而${\mathcalτ}: V(G)\rightarrow \Bbb{N}\cup \{0\}$是对$G$的顶点的阈值分配。如果$G$的顶点可以被划分成子集$D_0, D_1,..., D_k$，使得$D_0=D$和对于任何$i\in {0, ..., k-1}$，$D_{i+1}$中的每个顶点$v$在$D_0\cup ... \cup D_i$中至少具有$τ(v)$邻居，则顶点的子集$D$被称为对应于$(G, τ)$的动态独占。动态垄断实际上是在模仿社交网络中影响力的不可逆转的传播。本文首先利用平均阈值和图的阶数给出了动态垄断的最小规模的一个下界。同时，我们还得到了图的最小顶点覆盖的一个上界。当图$G$至少包含一个奇顶点时，我们得到了任意动态垄断的最小规模的上界$|G|/2$，其中任意顶点$v$的阈值设为$\lceil (deg(v)+1)/2 \rceil$（即严格多数阈值）.该界改进了严格多数阈值的最佳界。我们表明，后者的界限可以实现一个多项式时间算法。我们还证明了$α'(G)+1$是严格多数动态垄断规模的一个上界，其中$α'(G)$表示$G$的匹配数.最后，我们得到了一个基本上界的最小尺寸的任何动态垄断，在平均阈值和顶点度。使用这个界限，我们得到一些其他的上限。
英文摘要：
Let $G$ be a graph and ${\mathcalτ}: V(G)\rightarrow \Bbb{N}\cup \{0\}$ be an assignment of thresholds to the vertices of $G$. A subset of vertices $D$ is said to be a dynamic monopoly corresponding to $(G, τ)$ if the vertices of $G$ can be partitioned into subsets $D_0, D_1,..., D_k$ such that $D_0=D$ and for any $i\in {0, ..., k-1}$, each vertex $v$ in $D_{i+1}$ has at least $τ(v)$ neighbors in $D_0\cup ... \cup D_i$. Dynamic monopolies are in fact modeling the irreversible spread of influence in social networks. In this paper we first obtain a lower bound for the smallest size of any dynamic monopoly in terms of the average threshold and the order of graph. Also we obtain an upper bound in terms of the minimum vertex cover of graphs. Then we derive the upper bound $|G|/2$ for the smallest size of any dynamic monopoly when the graph $G$ contains at least one odd vertex, where the threshold of any vertex $v$ is set as $\lceil (deg(v)+1)/2 \rceil$ (i.e. strict majority threshold). This bound improves the best known bound for strict majority threshold. We show that the latter bound can be achieved by a polynomial time algorithm. We also show that $α'(G)+1$ is an upper bound for the size of strict majority dynamic monopoly, where $α'(G)$ stands for the matching number of $G$. Finally, we obtain a basic upper bound for the smallest size of any dynamic monopoly, in terms of the average threshold and vertex degrees. Using this bound we derive some other upper bounds.
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