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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2604.18949 |
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| _version_ | 1866910152670576640 |
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| author | Kim, Dohoon Woo, Eungyu Shin, Donghoon |
| author_facet | Kim, Dohoon Woo, Eungyu Shin, Donghoon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18949 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lions and Contamination: Trees and General Graphs Kim, Dohoon Woo, Eungyu Shin, Donghoon Combinatorics Computational Complexity 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. |
| title | Lions and Contamination: Trees and General Graphs |
| topic | Combinatorics Computational Complexity |
| url | https://arxiv.org/abs/2604.18949 |