AI 中文总结
研究图的僵尸伤害数\(\zdmg(G)\)，它通过要求追捕者沿最短路径追捕得到，衡量测地线追捕代价。证明\(\dmg(G)\leq\zdmg(G)\)，刻画\(\zdmg(G)=0\)的图，确定多种图的该参数值，还给出相关论证及界，显示其与普通伤害数的分离。
AI 中文摘要
在警察与强盗的伤害变体中，“伤害数”\(\dmg(G)\)是指在最优玩法下强盗所伤害的不同顶点的数量，由一名警察使其最小化而强盗使其最大化。我们引入“僵尸伤害数”\(\zdmg(G)\)，它是通过要求追捕者每一步都沿着最短路径朝向幸存者移动得到的。该参数衡量了以围堵而非单纯捕获为目标时测地线追捕的代价。我们证明了\(\dmg(G)\leq\zdmg(G)\)并刻画了\(\zdmg(G)=0\)的图。测地线追捕在树上不会造成额外伤害，我们确定了路径、圈、分裂图和完全多部图的该参数值。特别是，\(n\geq5\)时\(\zdmg(C_n)=n\)，当每个\(n_i\geq2\)时\(\zdmg(K_{n_1,\ldots,n_k})=n_1 + n_2 - 2\)。我们还为稀疏图开发了一种非回溯追踪论证。如果\(G\)是连通的，\(\delta(G)\geq2\)且\(g(G)\geq5\)，那么每个顶点都会被伤害，因此\(\zdmg(G)=n(G)\)。对于每个围长至少为五的连通图，同样的论证给出了尖锐的界\(\zdmg(G)\geq g(G)\)。由此可知，对最小度至少为二的连通图的每条边进行完全细分会产生具有最大僵尸伤害的图。这些结果与普通伤害数产生了无界的分离。导致这些结果的大多数问题以及开放猜想都是通过“理论猜想”产生的，这是一个由导师监督的发现循环，结合了自动猜想、精确计算、语言模型辅助探索和人类数学判断。
英文摘要
In the damage variant of Cops and Robber, the \emph{damage number} \(\dmg(G)\) is the number of distinct vertices damaged by the robber under optimal play, with one cop minimizing and the robber maximizing this number. We introduce the \emph{zombie damage number} \(\zdmg(G)\), obtained by requiring the pursuer to move at every turn along a shortest path toward the survivor. The parameter therefore measures the cost of geodesic pursuit when the objective is containment rather than capture alone. We prove that \(\dmg(G)\leq\zdmg(G)\) and characterize the graphs with \(\zdmg(G)=0\). Geodesic pursuit incurs no additional damage on trees, and we determine the parameter exactly for paths, cycles, split graphs, and complete multipartite graphs. In particular, \(\zdmg(C_n)=n\) for \(n\geq5\), while \(\zdmg(K_{n_1,\ldots,n_k})=n_1+n_2-2\) when \(n_i\geq2\) for every \(i\). We also develop a nonbacktracking trace argument for sparse graphs. If \(G\) is connected, \(δ(G)\geq2\), and \(g(G)\geq5\), then every vertex is damaged, and hence \(\zdmg(G)=n(G)\). The same argument gives the sharp bound \(\zdmg(G)\geq g(G)\) for every connected graph of finite girth at least five. It follows that fully subdividing each edge of a connected graph of minimum degree at least two produces a graph with maximum zombie damage. These results yield an unbounded separation from the ordinary damage number. Most of the questions leading to these results, as well as the open conjectures, arose through \textsc{Theo-Conjecture}, an advisor-supervised discovery loop combining automated conjecturing, exact computation, language-model-assisted exploration, and human mathematical judgment.
