离散时间治疗数
Discrete-time treatment number
AI总结：
本文提出图的离散时间治疗数，最小化每步最大治疗次数，证明其与路径宽度的上界关系，并给出网格、树等图类的精确结果。
摘要：
我们引入图的离散时间治疗数，其中每个顶点在任意给定时间步恰好处于三种状态之一：受损、脆弱或已治疗。我们的治疗数不同于仅使用两种状态的其他图搜索参数，例如消防员问题或Bernshteyn和Lee的检查数。顶点代表个体，边存在于关系密切的个体之间。每个顶点初始为受损状态；即使在治疗后，它也可能再次受损。我们的目标是治疗整个群体，使得在最后一个时间步，没有成员处于脆弱或受损状态，同时最小化每个时间步发生的最大治疗次数。这个最小值即为治疗数，它取决于预先确定的顶点可保持已治疗状态的时间长度$r$和顶点可保持脆弱状态而无需再次治疗的时间长度$s$的选择。我们用$pw(H)$表示图$H$的路径宽度，并证明$H$的治疗数上界为$\lceil \frac{1+pw(H)}{r+s}\rceil$。这等于谨慎治疗计划的最佳可能下界，谨慎治疗计划定义为每个顶点在首次治疗后，在随后的每个连续$r+s$时间步内再次接受治疗，直至其最后一次治疗。然而，许多图允许非谨慎的计划。当$r=s=1$时，我们找到了证明下界的有用工具，证明了$n\times n$网格的治疗数等于$\lceil\frac{1+n}{2}\rceil$，刻画了每个时间步仅需一次治疗的图，并证明细分一条边可以降低治疗数。已知存在路径宽度任意大的树；令人惊讶的是，我们证明对于任何树$T$，存在$T$的一个细分，使得每个时间步最多需要两次治疗。
英文摘要：
We introduce the discrete-time treatment number of a graph, in which each vertex is in exactly one of three states at any given time-step: compromised, vulnerable, or treated. Our treatment number is distinct from other graph searching parameters that use only two states, such as the firefighter problem or Bernshteyn and Lee's inspection number. Vertices represent individuals and edges exist between individuals with close connections. Each vertex starts out as compromised; it can become compromised again even after treatment. Our objective is to treat the entire population so that at the last time-step, no members are vulnerable or compromised, while minimizing the maximum number of treatments that occur at each time-step. This minimum is the treatment number, and it depends on the choice of a pre-determined length of time $r$ that a vertex can remain in a treated state and length of time $s$ that a vertex can remain in a vulnerable state without being treated again. We denote the pathwidth of graph $H$ by $pw(H)$ and prove that the treatment number of $H$ is bounded above by $\lceil \frac{1+pw(H)}{r+s}\rceil$. This equals the best possible lower bound for a cautious treatment plan, defined as one in which each vertex, after being treated for the first time, is treated again within every consecutive $r+s$ time-steps until its last treatment. However, many graphs admit a plan that is not cautious. When $r=s=1$, we find a useful tool for proving lower bounds, show that the treatment number of an $n\times n$ grid equals $\lceil\frac{1+n}{2}\rceil$, characterize graphs that require only one treatment per time-step, and prove that subdividing one edge can reduce the treatment number. It is known that there are trees with arbitrarily large pathwidth; surprisingly, we prove that for any tree $T$, there is a subdivision of $T$ that requires at most two treatments per time-step.
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。
