This paper investigates a special variant of a pursuit-evasion game called lions and contamination. In a graph where all vertices are initially contaminated, a set of lions traverses the graph, clearing the contamination from every vertex they visit. However, the contamination simultaneously spreads to any adjacent vertex not occupied by a lion. We analyze the relationships among the lion number $\mathcal{L}(G)$, monotone lion number $\mathcal{L}^m(G)$, and the graph's pathwidth $\operatorname{pw}(G)$. Our main results are as follows: (a) We prove a monotonicity property: for any graph $G$ and its isometric subgraph $H$, $\mathcal{L}(H)\le \mathcal{L}(G)$. (b) For trees $T$, we show that the lion number is tightly characterized by pathwidth, satisfying $\operatorname{pw}(T)\le \mathcal{L}(T)\le \operatorname{pw}(T)+1$. (c) We provide a counterexample showing that the monotonicity property fails for arbitrary subgraphs. (d) We show that, in contrast to the tree case, pathwidth does not yield a general lower bound on $\mathcal{L}(G)$ for arbitrary graphs. (e) For any connected graph $G$, we prove the general upper bound $\mathcal{L}(G)\le \operatorname{pw}(G)+1$. (f) For the monotone variant, we establish the general lower bound $\operatorname{pw}(G)\le \mathcal{L}^m(G)$. (g) Conversely, we show that $\mathcal{L}^m(G)\le 2\operatorname{pw}(G)+2$ holds for all connected graphs, which is best possible up to a small additive constant.

翻译：本文研究一种称为“狮子与污染”的追逃博弈的特殊变体。考虑一个所有顶点初始时均被污染的图，一组狮子在图上游走，清除所经顶点的污染。然而，污染会同时扩散至任意未被狮子占据的相邻顶点。我们分析了狮子数$\mathcal{L}(G)$、单调狮子数$\mathcal{L}^m(G)$与图的路径宽度$\operatorname{pw}(G)$之间的关系。主要结果如下：(a) 证明了一个单调性性质：对任意图$G$及其等距子图$H$，有$\mathcal{L}(H)\le \mathcal{L}(G)$。(b) 对于树$T$，表明狮子数由路径宽度紧刻画，满足$\operatorname{pw}(T)\le \mathcal{L}(T)\le \operatorname{pw}(T)+1$。(c) 给出了一个反例，表明该单调性性质对任意子图不成立。(d) 证明了与树的情况相反，路径宽度不构成一般图$G$上$\mathcal{L}(G)$的通用下界。(e) 对任意连通图$G$，证明了通用上界$\mathcal{L}(G)\le \operatorname{pw}(G)+1$。(f) 对于单调变体，建立了通用下界$\operatorname{pw}(G)\le \mathcal{L}^m(G)$。(g) 相反地，证明了所有连通图满足$\mathcal{L}^m(G)\le 2\operatorname{pw}(G)+2$，且该界除小附加常数外是最优的。