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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.05663 |
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| _version_ | 1866915233331675136 |
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| author | Chen, Rong |
| author_facet | Chen, Rong |
| contents | For any pair of edges $e,f$ of a graph $G$, we say that {\em $e,f$ are $P_3$-connected in $G$} if there exists a sequence of edges $e=e_0,e_1,\ldots, e_k=f$ such that $e_i$ and $e_{i+1}$ are two edges of an induced $3$-vertex path in $G$ for every $0\leq i\leq k-1$. If every pair of edges of $G$ are $P_3$-connected in $G$, then $G$ is {\em $P_3$-connected}. $P_3$-connectivity was first defined by Chudnovsky et al. in 2024 to prove that every connected graph not containing $P_5$ as an induced subgraph has cop number at most two. In this paper, we give a characterization of $P_3$-connected graphs and prove that a simple graph is $P_3$-connected if and only if it is connected and has no homogeneous set whose induced subgraph contains an edge. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05663 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterization of $P_3$-connected graphs Chen, Rong Combinatorics For any pair of edges $e,f$ of a graph $G$, we say that {\em $e,f$ are $P_3$-connected in $G$} if there exists a sequence of edges $e=e_0,e_1,\ldots, e_k=f$ such that $e_i$ and $e_{i+1}$ are two edges of an induced $3$-vertex path in $G$ for every $0\leq i\leq k-1$. If every pair of edges of $G$ are $P_3$-connected in $G$, then $G$ is {\em $P_3$-connected}. $P_3$-connectivity was first defined by Chudnovsky et al. in 2024 to prove that every connected graph not containing $P_5$ as an induced subgraph has cop number at most two. In this paper, we give a characterization of $P_3$-connected graphs and prove that a simple graph is $P_3$-connected if and only if it is connected and has no homogeneous set whose induced subgraph contains an edge. |
| title | Characterization of $P_3$-connected graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.05663 |