围长至少为5的连通图和树中度比例控制的动态垄断
Dynamic Monopolies for Degree Proportional Thresholds in Connected Graphs of Girth at least Five and Trees
摘要：
设$G$是一个图，令$ρ\in (0,1)$。对于$G$的顶点的集合$D$，让集合$H_ρ(D)$通过从集合$D$开始，并且迭代地将更多的顶点$u$添加到当前集合，如果它们在其中至少有$\lceil ρd_G(u)\rceil$邻居。如果$H_ρ(D)$包含$G$的所有顶点，则$D$被称为不可逆动态垄断或与阈值函数$u\mapsto \lceil ρd_G(u)\rceil$相关联的完美目标集合。让$h_ρ(G)$成为这种不可逆动态垄断的最小基数。对于最大度至少为$\frac{1}ρ$的连通图$G$，Chang（Triggering cascades on undirected connected graphs，Information Processing Letters 111（2011）973-978）示出了$h_ρ(G)\leq 5.83ρn(G)$，其由Chang和Lyuu（Triggering cascades on strongly connected directed graphs，Theoretical Computer Science 593（2015）62-69）改进为$h_ρ(G)\leq 4.92ρn(G)$。我们证明了对于每个$ε>0$，存在一些$ρ(ε)>0$使得对于$(0,ρ(ε))$中的每个$ρ$，有$h_ρ(G) \leq(2+ε)ρn(G)$，并且每个连通图$G$的最大度至少为$\frac{1}ρ$，围长至少为$5$。此外，我们表明，$h_ρ(T) \leq ρn(T)$的每一个$ρ$在$(0,1]$，和每一个树$T$的顺序至少$\frac{1}ρ$。
英文摘要：
Let $G$ be a graph, and let $ρ\in (0,1)$. For a set $D$ of vertices of $G$, let the set $H_ρ(D)$ arise by starting with the set $D$, and iteratively adding further vertices $u$ to the current set if they have at least $\lceil ρd_G(u)\rceil$ neighbors in it. If $H_ρ(D)$ contains all vertices of $G$, then $D$ is known as an irreversible dynamic monopoly or a perfect target set associated with the threshold function $u\mapsto \lceil ρd_G(u)\rceil$. Let $h_ρ(G)$ be the minimum cardinality of such an irreversible dynamic monopoly. For a connected graph $G$ of maximum degree at least $\frac{1}ρ$, Chang (Triggering cascades on undirected connected graphs, Information Processing Letters 111 (2011) 973-978) showed $h_ρ(G)\leq 5.83ρn(G)$, which was improved by Chang and Lyuu (Triggering cascades on strongly connected directed graphs, Theoretical Computer Science 593 (2015) 62-69) to $h_ρ(G)\leq 4.92ρn(G)$. We show that for every $ε>0$, there is some $ρ(ε)>0$ such that $h_ρ(G) \leq(2+ε)ρn(G)$ for every $ρ$ in $(0,ρ(ε))$, and every connected graph $G$ that has maximum degree at least $\frac{1}ρ$ and girth at least $5$. Furthermore, we show that $h_ρ(T) \leq ρn(T)$ for every $ρ$ in $(0,1]$, and every tree $T$ that has order at least $\frac{1}ρ$.
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。
