A Menger-type theorem for two induced paths
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929352860499968 |
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| author | Albrechtsen, Sandra Huynh, Tony Jacobs, Raphael W. Knappe, Paul Wollan, Paul |
| author_facet | Albrechtsen, Sandra Huynh, Tony Jacobs, Raphael W. Knappe, Paul Wollan, Paul |
| contents | We give an approximate Menger-type theorem for when a graph $G$ contains two $X-Y$ paths $P_1$ and $P_2$ such that $P_1 \cup P_2$ is an induced subgraph of $G$. More generally, we prove that there exists a function $f(d) \in O(d)$, such that for every graph $G$ and $X,Y \subseteq V(G)$, either there exist two $X-Y$ paths $P_1$ and $P_2$ such that the distance between $P_1$ and $P_2$ is at least $d$, or there exists $v \in V(G)$ such that the ball of radius $f(d)$ centered at $v$ intersects every $X-Y$ path. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_04721 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Menger-type theorem for two induced paths Albrechtsen, Sandra Huynh, Tony Jacobs, Raphael W. Knappe, Paul Wollan, Paul Combinatorics Discrete Mathematics 05C38, 90C27, 05C40, 05C12 We give an approximate Menger-type theorem for when a graph $G$ contains two $X-Y$ paths $P_1$ and $P_2$ such that $P_1 \cup P_2$ is an induced subgraph of $G$. More generally, we prove that there exists a function $f(d) \in O(d)$, such that for every graph $G$ and $X,Y \subseteq V(G)$, either there exist two $X-Y$ paths $P_1$ and $P_2$ such that the distance between $P_1$ and $P_2$ is at least $d$, or there exists $v \in V(G)$ such that the ball of radius $f(d)$ centered at $v$ intersects every $X-Y$ path. |
| title | A Menger-type theorem for two induced paths |
| topic | Combinatorics Discrete Mathematics 05C38, 90C27, 05C40, 05C12 |
| url | https://arxiv.org/abs/2305.04721 |